# Proof involving series of the powers of n

**URL:** <https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461>\
**Category:** Factual Questions\
**Created:** [November 30, 2016, 4:03pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461 "2016-11-30T16:03:10Z")\
**Posts on this page:** 12\
**Page:** 2

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**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\
**Post date:** [December 1, 2016, 2:13pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/21 "2016-12-01T14:13:35Z")

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> [@Indistinguishable](#):
>
> Frankly, I hate it when a layperson someone stumbles upon something genuine and interesting and lovely in math all on their own, as happened in the OP, and everyone rushes to tell them how they are wrong and their discovery worthless because it doesn’t conform to the perspective (often, simply the formal language) the scolders have erroneously taken from their overly rigid textbooks to be the only useful way to talk or think.
> 
> Or, rather, I love the first part and hate the second part.

Agreed, and I hope I wasn’t guilty of this. The OP asked “Did I do this right?”; and neither an unqualified Yes nor an unqualified No would have been an appropriate answer.

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 1, 2016, 2:45pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/22 "2016-12-01T14:45:17Z")

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**Thudlow** , no, I didn’t perceive you shooting me down at all! Over the years, you’ve consistently answered many of my math questions in a way that I can understand, and so I knew where you were coming from. I know from seeing answers in areas where I have a certain level of expertise (art, music and marketing), that there often are nuances at play where as you put it “neither an unqualified yes nor an unqualified no would have been the appropriate answer”

**Indistinguishable** , thank you for comments and perspective. I remember reading your post on the various means of summation in the “sum of natural numbers is -1/12” thread and found that very helpful.

So let’s go down the rabbit hole further, shall we? I thought of this as I was falling asleep last night. Apologies, because I don’t know how to express this robustly, but:

Given a Ramanujan summation of the series of the powers of n, n\>1 (although I noted **Indistinguishable’s** point about it works for n\>0 and \< 1 but don’t fully understand that yet), we get the following:

n=2, S = -1  
n=3, S = -1/2  
n=4, S = -1/3  
n=5, S = -1/4  
n=6, S = -1/5

So, ladies and gentlemen, that’s the harmonic series in a negative direction. As even I know, the harmonic series is divergent, and so adds to negative infinity (by an intuitive method of summation). So what that means is by this summation method, when you add a shitload of non-prime, positive integers together, you get negative infinity.

Jesus Christ, that’s amazing! Assuming one of youse don’t come in here and tell me I made (yet another) mistake. 😉

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [December 1, 2016, 9:30pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/23 "2016-12-01T21:30:24Z")

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> [@Thudlow\_Boink](#):
>
> Agreed, and I hope I wasn’t guilty of this.

You’re alright; my ire was focused elsewhere. Your post #12 put it fine:

> [@Thudlow\_Boink](#):
>
> I wouldn’t exactly say that it’s _wrong_, just that it’s based on assumptions that aren’t always warranted, so that your result isn’t universally valid.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [December 1, 2016, 10:15pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/24 "2016-12-01T22:15:27Z")

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> [@D18](#):
>
> Given a Ramanujan summation of the series of the powers of n, n\>1 (although I noted **Indistinguishable’s** point about it works for n\>0 and \< 1 but don’t fully understand that yet), we get the following:
> 
> n=2, S = -1  
> n=3, S = -1/2  
> n=4, S = -1/3  
> n=5, S = -1/4  
> n=6, S = -1/5
> 
> So, ladies and gentlemen, that’s the harmonic series in a negative direction. As even I know, the harmonic series is divergent, and so adds to negative infinity (by an intuitive method of summation). So what that means is by this summation method, when you add a shitload of non-prime, positive integers together, you get negative infinity.
> 
> Jesus Christ, that’s amazing! Assuming one of youse don’t come in here and tell me I made (yet another) mistake. 😉

Yes, that’s all correct. In the same sense that 1 + 2 + 4 + 8 + … = -1, that 1 + 10 + 100 + 1000 + … = -1/9, and so on, we may say that the sum of n^k over all n \>= 2 and k \>= 0 is -∞.

Of course, there is another, more straightforward sense in which 1 + 2 + 4 + 8 + … = +∞, and in that sense, the sum of n^k over all n \>= 2 and k \>= 0 is +∞ as well. In this more straightforward sense, adding a lot of increasingly large positive values gives a positively infinite result, not a negatively infinite result. Math doesn’t deny this and no one is wrong to say this; there are just multiple different perspectives available to us.

* * *

You may find it interesting to consider now the sum n + 2n^2 + 3n^3 + 4n^4 + …. Can you figure out what this should come out to?

[And once you’ve done that, you may find it interesting to consider the sum of (k + 1) n^k over all n \>= 2 and k \>= 0. But that’s rather a challenge…]

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 2, 2016, 2:56pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/25 "2016-12-02T14:56:54Z")

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Cool! I hit a roadblock on n + 2n^2 + 3n^3 + 4n^4 + …., but when I did n^0 + n + 2n^2 + 3n^3 + 4n^4 + …., I got it back to -1/(1-n), so the positive of n^0+n^1+n^2. Which is pretty cool, if correct. Is that what you had in mind?

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [December 2, 2016, 8:24pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/26 "2016-12-02T20:24:26Z")

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You got that 1 + n + 2n^2 + 3n^3 + 4n^4 + … = -1/(1-n)? That doesn’t seem correct; for example, at n = 0, the left-hand side is 1 and the right-hand side is -1. What was your reasoning?

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 3, 2016, 1:34am UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/27 "2016-12-03T01:34:34Z")

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OK, you’re right. Made a mistake in the subtraction. (However, isn’t the implication n \> 1, here?)

I just played around with it again, and think I have the answer, but want to sleep on it and check my math in the morning. This is fun!

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 3, 2016, 3:04pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/28 "2016-12-03T15:04:08Z")

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Before I answer this, can you please confirm that I have this correct. Given:

S1 = n1 + n2 + n3 + …

S2 = a1 + a2 + a3 + …

That it is cool to subtract them in this way:

S1 - S2 = n1 + (n2 - a1) + (n3 - a2) + …

Tony in the infamous Numberphile video of the natural numbers = -1/12 did this in one step of his proof, but he was adding. So I just wanted to make sure that this is right and holy with subtracting.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [December 3, 2016, 6:24pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/29 "2016-12-03T18:24:42Z")

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For the notions of summation invoked in this thread, yeah, that would be fine. That’s certainly how I would happily expect you to be reasoning about these series.

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 3, 2016, 6:48pm UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/30 "2016-12-03T18:48:39Z")

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Very good!

Cecil help me, here goes:

S = n + 2n^2 + 3n^3 + 4n^4 + …

nS = n^2 + 2n^3 + 3n^4 + 4n^5 + …

S – nS = n + (2n^2 – n^2) + (3n^3 – 2n^3) + (4n^4 – 3n^4) + …

(1-n)S = n + n^2 + n^3 + n^4 + …

(1-n)S = n (1 + n + n^2 + n^3 + …)

Using the sum demonstrated up-thread for 1 + n + n^2 + n^3 + …

(1-n)S = n (1/(1-n))

S = n

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<div class="post-metadata">

**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [December 4, 2016, 1:52am UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/31 "2016-12-04T01:52:27Z")

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> [@D18](#):
>
> (1-n)S = n (1/(1-n))

Perfect all the way up through this penultimate line!

> [@](#):
>
> S = n

Whoops, you made a slight slip in the algebra moving from the previous line to this one.

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**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\
**Post date:** [December 4, 2016, 3:07am UTC](https://boards.straightdope.com/t/proof-involving-series-of-the-powers-of-n/773461/32 "2016-12-04T03:07:47Z")

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:smack:

S = n/(1-n)^2

It is awesome how these series can turn out! I keep thinking “Oh my God, it’s full of stars!”

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