Sum of alternating harmonic series

Question

Edit: As @FamousBlueRaincoat pointed out below, this question was based on an error in a wikipedia article.

The Wikipedia article on the harmonic series gives the following "proof without words" that the alternating harmonic series $1-1/2 +1/3 -1/4 + \cdots $ converges to $\log (2) $: \begin{equation} (1/1)(1/1 - 1/2) + (1/2)(2/3 -2/4) + (1/4)(4/5 - 4/6 + 4/7 - 4/8) + \cdots = \log (2). \end{equation}

Can anyone explain why the sum on the left is $\log (2) $? I don't see it right now.

Edit: my goal is specifically to understand this "proof without words" that the alternating harmonic series converges to $\log(2)$.

Why write $(1/1)(1/1 - 1/2) + (1/2)(2/3 -2/4) + (1/4)(4/5 - 4/6 + 4/7 - 4/8)$ instead of $1-1/2 +1/3 -1/4 + ...$? They are equal quantities and I don't yet see an advantage to writing it the former way.. — graydad, Feb 03 '15 at 19:17
possible duplicate of Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ — Dan, Feb 04 '15 at 22:22
@LarryFreeman That answer does not explain the proof quoted above. — , Feb 05 '15 at 00:19
The answer to your question is that Wikipedia article should be edited. The quoted manipulation does not demonstrate anything on its own. By following the link to the source you will see a geometric argument in which areas of certain rectangles correspond to the terms listed above. But without the picture, the series written in your post is just a pointless perturbation of the original problem. — , Feb 05 '15 at 00:21

score 3 · Answer 1 · answered Feb 03 '15 at 19:19

3

For $|x|<1$,

$$\frac 1{1+x}=\sum_{n=0}^\infty (-1)^nx^n$$

Integrate both sides to get $$\log(1+x)=C+\sum_{n=0}{(-1)^nx^{n+1}\over n+1}$$

It's not hard to see that $C=0$, and reindexing gives

$$\log(1+x)=\sum_{n=1}^\infty{(-1)^{n-1}x^n\over n}$$

Now, (waving of hands) let $x\to 1$ on both sides to get the result.

answered Feb 03 '15 at 19:19

Tim Raczkowski

20,046

Thank you, I'm specifically trying to understand the "proof without words" though. (I believe Abel's theorem takes care of the hand-waving part of the argument you gave.) – littleO Feb 03 '15 at 19:27
Oh, I see. Yes, you're right about Abel's theorem. – Tim Raczkowski Feb 03 '15 at 19:28

Sum of alternating harmonic series

1 Answers1