Let's take another look at Infinite Series et al.

This article will take a look, at Infinite Series, with a special emphasis towards the interesting conundrum arising out of the Ramanujan Summation & Zeta function regularization.

Now to begin with, as a prologue... The Summation / Sum of an Infinite Series is defined / exits : if the Partial Sums of that Infinite Series either approaches a finite value (thus resulting in a Convergent series), or the Partial Sums of that Infinite Series goes to ± infinity (resulting in a Divergent series).

Two quick examples whould elaborate, further solidify the meaning of above statements ...

Example 1:

Take the series : 

1/2 + 1/4 + 1/8  + 1/16 + .... + 1/2ⁿ

Now the sum of above series can be shown to approach 1, as n approaches infinity.

One can refer to the many articles / videos on the internet, that clearly illustrates geometrically, as to why the summation of above series approaches 1. (Hint: 1/2 of a square + 1/2 of the remaining square + 1/2 of that remainder + etc etc = full square)

Thus: Σ 1/2ⁿ ⟶ 1 (as n ⟶ ∞)

Thus the above is an example of a convergent series (aka the partial sums of a series upto the n^th term approaches / converges to a finite value).

Example 2:

The series : 1 + 2 + 3 + 4 + ... + n , is a divergent series, because when we increase the value of n, we observe that the partial sums (upto n), keeps on increasing, as n increases. Thus when n approaches infinity, the sum of the series as well whould approach  positive infinity.

Thus: Σ n ⟶  +∞ (as n ⟶ ∞)

And thus 1 + 1 + 1 + 1 +... +1ⁿ, is a divergent series.

But then something interesting happens, when in above series, one replaces each subsequent plus (+) with a minus (-).

Aka, take the following series...

1 - 1 + 1 - 1 +... (-1)ⁿ  ( for n >= 0 )

Now observe that when n is an even number, the partial sum upto n, will be 1.

But when n is an odd number, the partial sum upto n, will be 0.

Another frequently used technique is to use parenthesis to group the above series as in...

(1 - 1) + (1 - 1) +... = 0

But...

1 + (- 1 + 1) + (- 1 + 1)  + ... = 1

If so, what whould be the sum of the above series when n ⟶ ∞ ? You see, when we mathematically define ∞, there is no notion / concept of assigning an odd / even notation to ∞. For mathematically ∞, is just a number that is beyond the reach of all countable numbers (and that's it...) If it's uncountable, then how can we even say / state / know as to whether it is even or odd ?

So this is the first Conumdrum... 

Well sort of... 

Because if one looks at the modern / current definition of the Sum of an Infinite Series... One observes it clearly mentioning that for a Sum of a Series to exist / be defined, the Partial Sums needs to either converge or needs to go to ± infinity.

But the series: Σ (-1)ⁿ ( for n >= 0 ), satisfies neither of the above two conditions.

THUS...

Well, you may think, that via a THUS clause, this Conumdrum whould now be / get resolved. And all mathematicians whould peacefully retreat / go back home, have dinner and go to sleep (and never take up this matter ever ever again...)

Well you see (as in any other field), even in the field of mathematics, one finds types of the class Adamant / Obstinate / Dogged / Persistent  (select all answers that are correct...). For if someone came to me and shows me the definition of the Sum of... and produces a THUS clause on me... at the minimum I whould give him / her a look of / a glance of... (aka : THUS I Agree to Disagree... I THUS respond with a contraTHUS... 😁)

You see... Who came up with the definition of Sum of...? It was yet another... 

THUS... The definitions themselves are eventually rooted in the Base Axioms / Postulates of mathematics. And next, based / rooted on these Base Axioms, we next have Definitions, Corollary(s) etc. etc.

THUS, many mathematicians when confronted with a Conundrum as presented in the example of the Sum of... see it as a... 

[ Long Side Note : 

There were / are / will be millions / tens of millions... whom engages in mathematics as a recreational activity as well... - aka Recreational Mathematics- aka they peruse mathematics and engage in analytical activities for fun! I wonder as to whether I have ever met someone like that... hmmm 🤔 !?

Thus some such adventurous individuals whould take a deeper dive into the depths of the jungles of... , with hopes of / fingers crossed to, finding the foundations of an Eldorado or... 

And many have indeed found many many interesting foundations hitherto hidden in the jungles of... Most all / everything we study in mathematics, physics etc. etc. are the results of such ventures...

Of course occasionally some adventurers, whom boldly have had so ventured into the jungles of... do have come across / being tackled to the ground by "unforseen circumstances / obstacles /... "

THUS : All, whom boldly ventures into "the field", are kindly reminded to practice at least the minimum precautions when... you know... physical / psychological / etc... Thus just like in any other field, even in the field of the sciences, one observes that probably / maybe about 10%  to develop serious issues, out of / resulting from / due to their deep dives and / or turbulant interactions with mother nature... 

Three examples : 

1. Ignaz Semmelweis (1818 – 1865). Got locked up in an asylum, because his colleagues despised him, for they were certain that there cannot be entities not visible to the naked eyes. (Of course today we call such entities bacteria / viruses etc).

 https://en.m.wikipedia.org/wiki/Ignaz_Semmelweis

2. Georg Cantor (1845 – 1918). Mathematician who played the pivotal role in the formation of the Set Theory. Multiple sources, whould provide varied views as to why Georg Cantor eventually developed spycholigical problems. One line of thought, is that it was due to Heavy Criticism of his work, from some of his contemporaries, and then the death if his youngest son etc all compounding into / leading into severe depression. The other line of thought is that, he mentally overstrained himself, for example with / via the such of : The works on Continuum Hypothesis.

https://en.m.wikipedia.org/wiki/Georg_Cantor

3. Ludwig Boltzmann (1844 – 1906). Today considered as a pioneer in the fields of Thermodynamics etc. For example, the currently used Enthropy equation (the Boltzmann equation) :  S = k . ln Ω is one of his brain child. But due to some clashes of opinions, with some academics of his day, he later developed chronic depression.

He died by suicide on 5 September 1906, by hanging himself while on vacation...

https://en.m.wikipedia.org/wiki/Ludwig_Boltzmann

This may not be the most proper location, to bring this up... But an individual whom has the capacity to communicate with / see the Spirit World entities or Purgatory (that does not exist...👻 I know,  I know... 😊), once told the author of this text, that for a vast majority of us, it whould be better to somehow stay on earth 🏞️, rather than going to the "other side" (that does not exist ...👻). What he was implying was... if all / everything has failed to one -on planet earth-, well give a try to a sedentary / religious life 🕉️☸️☮️☯️☪️✝️☦️🕎(rather than...👿). One can do / commit to a sedentary life, even if one is in a total state of destitution... That way at least, one will be accruing a developed conscience / state of mind, and thus when the time comes (mother nature decides to...), no worries...💐

Hope I did not anger anyone too much, by bringing a topic, related to ethics & religion (which unfortunately via circumstances, is an out of bounds topic, to the editor of this text... don't ask...🤫🤐)

Let's take another look at the circumstances behind the three above sample cases...

1: Now when one takes a cursory look at the case of Ignaz Semmelweis, it's apparent that most likely / most probably the "lion's share" of the blame / fault, whould be / could be assigned to / towards his colleagues. One observes that in the field of Personnel Management, when one studies / analyses the "stack / file of complaints"... that with around probably 50% of the cases, the actual fault is found NOT to be with the individual(s) against whom the complaint(s) has / have been filed, but rather with the individual(s) whom has / have filed the original complaint(s). Thus in such cases, most likely, the person(s) whom filed the original complaint(s) must have had some dispute / clash of interest(s) / personal dislike / grudge / envy etc etc against / with the person(s) he / she / them filed the complaint(s)...

A modern day equivalent could be heavy Internet Bullying -via / through social media- and / or heavy organized slander campaigns... One even observes such in political campaigns, where each side have seperate teams / consultants to address / deal with such... 

But when individuals, are / get subjected onto such (as most individuals do not have teams of consultants working for them...), the end results sometimes are...

Thus in most likelihood in the case of Ignaz Semmelweis, the fault / the guilty party is most likely... (I will leave the "derivation" to the reader...)

2. The case of Georg Cantor, has been included into this discussion, specifically to highlight two points. 

2.1: If for the purpose of this discussion we assume, that his works caused him to overstrain himself... Then this whould a good point to highlight, the need to take care of ones physical / psychological /... well being... Thus if one feels either physically or psychologically strained, probably it whould be advisable to take a break... / relax a bit...? 

If for example you engage in a lot of physical activities / trainings, and at the end of the day your whole body aches... The reason for most of the panes could be the accumulation of lactic acid in the muscle tissues... A solution used by athletes for example, it to soak the affected regions / muscles, in cold water / ice. Cold water / ice, seems to either neutralise the lactic acids or numb / neutralise the pains... There are also special suits one can wear, which have ducts / tubings, through which cold water can be circulated, thus...

But regarding the usage of cold water, be however be attentive to do it carefully... as excessive exposure to cold, can sometimes lead to development of Colds / Flem / Pulmonary disorders / Pneumonia etc... 

This is why when we develop a Cold, we are advised to drink hot liquids, keep body warm etc. Observe that when we get a Cold / Flu, after a bout /episode of high fever / elevated body temperature, we usually feel a bit better. The reason is that, higher temperatures, kills many viruses / bacterias (as well as some amounts of body cells). But usually as there are much more body cells than viruses, the nett effect is usually / in most cases positive... But if fever goes beyond certain levels, immediate medical interventions may be required, because of the reasons described above...

Now back to our original discussion...

2.2: One observes that Georg Cantor seems to have had taken most of the Criticism quite personally, thus causing him to slip into depression? Same has happened when his youngest son died... 

If so, if that is the case, then a higher percentage of the fault, whould have to be / get assigned to / with Georg Cantor ? (What do you think... What is your opinion...?).

Now for example one observes, that even the likes of Albert Einstein and Neils Bhor, to have had serious disagreements, for example with the such of  Quantum Entanglement Phenomenon...

Niels Bohr and Albert Einstein, pictured above had a long-running collegial dispute about what Quantum Mechanics implies...

(Source:https://en.m.wikipedia.org/wiki/Copenhagen_interpretation)

Thus now, the question arrises : What does one do / how to approach / deal with Negative Reviews or Counter Arguments to one's Opinions or One's Works...?

Now for example in the case of Niels Bohr and Albert Einstein, one observes that they were good friends / buddies. But in the case of Leopold Kronecker vs. Georg Cantor... it's a different story... (The case of Leopold Kronecker vs. Georg Cantor... is also a classic example, where an individual with higher authority / higher in the ladder can, if so required crush / subdue / burry / overide... aka. politics ... 🤐 ...😶)

Here we have to understand that Reviews / Opinions can be / fall into:

1: Constructive Criticism (in support of / positive feedbacks etc OR in disagreement / pointing out errors / faults / contradictions /  inconsistencies etc.)

2: Malicious Negative Criticism

3: Personal Views / Biases...

Note: Constructive Criticism, the likes of peer reviews are (on many occasions), very / highly valuable, to polish out a product (removal of glitches etc), to validate (or negate) a formulation and come out with a more robust / better end result.

THUS...

(The topic if left open for discussion... No Malicious comments please... 🤗)

3. Now in the case of Ludwig Boltzmann, we clearly observe that work related clashes (toxic work environment, due to clash of ideals / views, with some colleagues) had had led him to veer into Chronic Depression, leading him to taking his own life. 

The author of this article, previously brought up the point, that there whould be much more constructive avenues to resolve such problems, rather than decide to go to the "other side". 

No pun intended... and no kidding... What guarantees do you have, that on the "other side", the "pains in the neck", don't / won't exit...? Just think about it... Those whom dislikes us, when they die, where do you think they whould go to / or where whould they as well come to ? !!!😐!!! 

-You are also here...?!

-I came here first...!!

Thus if one, gets subjected to Tons of Hate OR ...

Whenever the author of this text hears of a case of suicide, it sort of brings a level of "disappointed anger"... 😖 

Now let's say one totally messed up, and now one cannot any more face one's family / friends / society etc etc.

What should one do...?

Or let's say, one got financially totally wreaked...

What should one do...?

Or let's say, one has a large number of "friends / well wishers" 🙃, whom first thing each morning, check whether you are still alive...!

What should one do...?

Or let's say, the stars were not properly aligned when one was born. (Welcome to the club...😁)

What should one do...?

Well one solution, might be to go and give a courtesy call to that Aunt you last met, when you were in grade three (or was it grade four...) ?

Hie Auntie... How are you doing? Can I hide in your cellar for a few years...? Oh it's already occupied is it? Ok bye.

Another solution, whould be as suggested previously, is to give a try to a sendentary / religious life...

You see... one doesn't necessarily / even have to join a particular order or monastery etc, to practice a path of spirituality... Neither one has to  necessarily even spend money... For example, say one silently wishes (in mind), to all / everyone one meets °have a good day...° Now each time one so wishes, that accrues / counts as an act of kindness...

Of course such have to be done, with a heart of compassion. Thus for example take the following scenarios, where one's thought is , °have a good day...° + status of the heart & facial expressions.

Thought		Heart		Face		Notes....
°have a good day...°		💝		🙂		✅ 👍
°have a good day...°		💝		🤗		✅ 👍
°have a good day...°		💛		😌		✅ 👍
°have a good day...°		🤍		😟		❌ 👎
°have a good day...°		🖤		😠		❌ 👎
°have a good day...°		🤍		😏		❌ 👎

The following will further illustrate / solidify this point...

Thought		Heart		Face		Notes....
°#$!@🐕🐷🐃🦨🐒...°		🖤		🤬		❌ 👎
°#$!@🐕🐷🐃🦨🐒...°		🖤		😡		❌ 👎
°#$!@🐕🐷🐃🦨🐒...°		🖤		🙂		❌ 👎

To do... Compare between Chronic Depression and episodes of "Low Blood Presure" in the mornings...🙂

THUS...

End of Long Side Note: ] 

Now to continue our original discussion of the infinite series... 🤗

Refer the internet for:

1. Grandi's series : https://en.m.wikipedia.org/wiki/Grandi%27s_series

2. Harmonic series (mathematics) : https://en.m.wikipedia.org/wiki/Harmonic_series_(mathematics)

Part I

Thus / Now after the above long multiple detours (lunch break...), let's get some work done, shall we...

Thus as we were saying (prior to the lunch break)... thousands, tens of thousands of mathematicians over the ages, have again and yet again taken dives / deep dives around this Sum of... conundrums 

Now if we return to the Sum of :

  Σ (-1)ⁿ ( for n >= 0 )

Now if you did go through the articles of Grandi's series, you would have observed that via multiple techniques mathematicians have also come up with the solution that:

S =   Σ (-1)ⁿ  = 1/2

Aka for example if

 S = 1 − 1 + 1 − 1 + 1 - ...

Then 

- S =  − 1 + 1 − 1 + 1 - 1 +... 

Next add 1 to both sides

1 + (- S) = 1 + (− 1 + 1 − 1 + 1 - 1 ...)

1 - S       = 1 − 1 + 1 − 1 + 1 - ...

Now on right hand side we again get the infinite series S (the one we started with).

Thus 

1 - S  = S

Thus 

2S = 1

And presto 

S = 1/2 

Thus...

S      = 1 - 1 + 1 - 1 + 1 - 1 +... = 1/2

What do you say to that ! 😯😁

Thus one observes two contradictory results...

If we use the Definition of Infinite Series: Then the series 1 − 1 + 1 − 1 + ... has no sum (sum is undefined).

But if we use alternate techniques : Then the series 1 − 1 + 1 − 1 + ... get  summed to = 1/2.

As pointed out in the Wikipedia page, these contradictory results had had fueled; what has been characterized as an "endless" and "violent" dispute between mathematicians.

The depth of this contradiction, has now reached such levels, that the above 1/2, types of derivations /  results are now even embeded in the such of String Theory as well (via the Ramanujan Summation)... And wait for it, wait for it... Such results have even been proven to be correct / consistent with some experimental results (for example Casimir Effect forces at the atomic / quantum levels !).

THUS, at where we are right now, we cannot any longer, reject the alternate solution (some brave mathematicians, have bodly and daringly defended even at great costs to their own... did I get too dramatic here...)

If one is interested, one could also peruse how the same above types of  techniques have been used by the such of Ramanujan to derive the now famous equation:

The following, is loosely based on, the derivation found on a notebook of Ramanujan.

Sn            = 1 + 2 + 3 + 4 + 5 +...

Thus... 

4Sn          = 4 + 8 + 12 + 16 +...

Next, we subtract 4Sn from alternate terms of Sn (in the following manner)

     Sn        = 1 + 2 + 3 + 4 + 5 + 6  +..
 - 4Sn               - 4        - 8        - 12 +...

   -3Sn       = 1 - 2 + 3 - 4 +  5  - 6  + ...

Next, let's consider the above orange color series (and let's name it Se)

Se   = 1 - 2 + 3 - 4 + 5 - 6 + ...

Next we add Se to Se, with one right shift...

  Se = 1 - 2 + 3 - 4 + 5 - 6 + ... 
+Se       +1  - 2 + 3 - 4 + 5 - 6 + ...

2Se  = 1 - 1 + 1 - 1 + 1 - 1 +...  (-->eq1)

But we had previously proven that

S      = 1 - 1 + 1 - 1 + 1 - 1 +... = 1/2

Thus eq1 yields to ...

2Se  = 1/2

aka

Se    = 1/4

Thus, sum of the orange series...

Se  = 1 - 2 + 3 - 4 + 5 - 6 +... = 1/4

Next, recall that at the beginning, we derived that...

-3Sn = 1 - 2 + 3 - 4 + 5 - 6 + ... 

 Thus now we have...   

 -3Sn = Se 

          = 1/4

Thus via simple arithmetics...

-3Sn  = 1/4

   Sn   = -1/12

Thus...

Sn = 1 + 2 + 3 + 4 + 5 +... = -1/12 

Yet another derivation of the same...

Sn  = 1 + 2 + 3 + 4 + 5 +...

Se   = 1 - 2 + 3 - 4 + 5  +...

Se - Sn = - 4 - 8 - 12 - 16...

              = -4 (1 + 2 + 3 + 4 + 5 +...)

              = -4 Sn

-3Sn      = Se                ( ---> eq1)

Next add Se to Se with one right shift

  Se    = 1 - 2 + 3 - 4 + 5 - 6 +... 

+Se          +1  - 2 +3  - 4 +5  - 6 +...

 2Se   = 1 - 1 + 1 - 1 + 1 - 1 +... = 1/2

   Se   = 1/4 

Thus via substituting in eq1

We get...

Sn = 1 + 2 + 3 + 4 + 5 +... = -1/12

One whould find the above Ramanujan summation value, even embeded in the formulaes of String Theory... And to add further "glamor to the fireworks", those String Theory formulaes have even been proven / validated, via the experimental results / measurements of Casimir Effect Forces...

[For a more mathematically rigorous proof of above result, one apparently has to use Reiman Series (Zeta function regularization via Analytic Continuation), which the author of this article is yet not fully familiar with / has not yet studied sufficiently...]

Part II

Next lets take a look at the "other side / other half" OR the Conventional Derivation, of the Sum of the Natural Numbers...

One of the most commonly used technique (to find the sum of Integers from 1 to n), is the following.

Let's assume that the Sum of the integers from 1 to n comes up to / adds up to the value S.

Thus in Ascending order:

1 + 2 + 3 + 4 + 5 + ... + (n-1) + n = S

Next if we add the same numbers in reverse order, then again we should also get the same sum of S. 

(For example if we add 1 + 2 + 3 + 4 + 5, we get the same value as when adding 5 + 4 + 3 + 2 + 1. Aka the order of addition has no effect on the final value / result).

Thus in Descending order: 

 n + (n-1) + ... + 5 + 4 + 3 + 2 + 1 = S

Now let's write the above two addition one next to the other...

Asc.   Des.

1         n
2         n-1
3         n-2
4         n-3
5         n-4
 :           :
 :           :
n-2       3
n-1       2
n          1

____    ____

  S         S

Next, let's add each row of above two columns (aka do a row wise addition).

Asc.   Des.   Asc. + Des.

1    +   n        =     n+1
2    +   n-1     =     n+1
3    +   n-2     =     n+1
4    +   n-3     =     n+1
5    +   n-4     =     n+1
 :           :
 :           :
n-2  +   3      =      n+1
n-1  +   2      =      n+1
n     +   1      =      n+1

____    ____          _____

S      +   S      =       2S

I will let the reader conclude /deduct as to why in / via the above : 

2S = n.(n+1)

Thus :

S = 1 + 2 + 3 + 4 + 5 + ... + n = n.(n+1) / 2

Thus as n increases upto infinity, above Sum should also approach / go to infinity...

However in Part I we deducted that:

S = 1 + 2 + 3 + 4 + 5 +... ∞ = -1/12

Thus do you see the contradiction / the conundrum... And mathematicians have been debating fiercely over this for...

Now for example if one goes to a shop and gets 1 + 2 + 3 + 4 + 5 +... of an item and then goes to the counter and shows the Ramanujan Summation Result, to the cashier and asks him / her to pay you 1/12 times the price of the item you brought... I do not think it's going to go too well...

Thus in the physical world, what applies / what is valid, is the Standard Summation...

Viz : S = 1 + 2 + 3 + 4 + 5 + ... + n = n(n+1) / 2

But as already mentioned previously, the Ramanujan Summation does seem to hold in areas the such of String Theory (aka sub atomic levels / realms).

Another interesting fact that mathematicians have observed is: 

The Standard Summation Result S = n(n+1) / 2, is a curve of the form y = x² + x. 

Aka S = ½ (n² + n).

Now the Integral value of the above curve from 0 to -1 has been shown to be = -1/12.

Thus  _-1∫⁰  ½ (n² + n) = -1/12

Is this just a coincidence or... One clearly observes the involvement of -1 in the above integration.

If one did remember, in the process of deriving the Ramanujan Sum, we did heavily use the Grandi's Series...

S = 1 - 1 + 1 - 1 + 1 - 1 +... = 1/2

Again we observe the alternating -1 appearing here as well... 🤔

Observe that the Grandi's Series is a Step Function (with infinite steps). One could also view it as an Oscillating Step Function (oscillating between +1 and -1). Very interesting indeed... We do get these types of quantum / step behaviors at sub atomic levels as well.  hmmm...🤔🧐 Most interesting indeed...

Now what could all these mean !? 🧐 Where does all these lead us to...

The importance of the above -1, has to be reiterated, re-re-emphasised for it plays a crucial role in this discussion.

Otherwise for example take the following Summation of the Integers.

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 + ...

Next lets group the above in sets of 3s, from one onwards...

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 + ...

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10 +

   = 1 +      9         +     18       +       27       + ...

S = 1 + 9 + 18 + 27 + 36 + 45 + ...

S = 1 + 9 ( 1 + 2 + 3 + 4 + 5 + ...)

Thus :

S = 1 + 9 ( S )   ----> eq. problmtik

Thus :

-8S = 1

 S = -1/8

But any individual whom has studied mathematics, to certain level whould be quick to point out the err / fault of the above derivation...

Hint: It's regarding Divergent Series... Aka the Sum of all of the integers is actually : S = ∞

Thus in  eq. problmtik of above

S = 1 + 9 ( S ) 

∞ = 1 + 9( ∞ )

     = 1 + ∞

Aka:

 ∞  =  ∞

And NOT S = -1/8 !

This point was brought up, to further highlight, the crucial factor / crucial role the -1 s of the Grandi"s Series (in this regard)... They come into effect / plays the vital role with / in preventing the Series from Diverging into Infinity...

Thus as such, the Derivations of the Ramanujan Summation, used in this article do have certain "technical faults". But for simplicity sake, the easier Derivations were employed. ( for demonstrational purposes).  More Robust / Mathematically correct Derivations of the Said Summation do exist... [To Do ... 😁]

[To be continued...]

Ly De Sandaru.

Sketchpad...

Α α, Β β, Γ γ, Δ δ, Ε ε, Ζ ζ, Η η, Θ θ, Ι ι, Κ κ, Λ λ, Μ μ, Ν ν, Ξ ξ, Ο ο, Π π, Ρ ρ, Σ σ/ς, Τ τ, Υ υ, Φ φ, Χ χ, Ψ ψ, Ω ω.