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I am trying to determine if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges using an alternating series test. The test in question requires me to prove $ 1 - n \log \left( \frac{n + 1}{n} \right) $ is decreasing and that $ \lim_{n \to \infty} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) = 0 $ to prove this series is convergent.

My intutition says that this series is convergent because $ n \log \left( \frac{n + 1}{n} \right) $ will tend towards 1 as n goes to infinity (due to the definition of e). I am having trouble proving the sequence $ 1 - n \log \left( \frac{n + 1}{n} \right) $ is decreasing.

I set up the inequality $ 1 - (n + 1) \log \left( \frac{n + 2}{n + 1} \right) \leq 1 - n \log \left( \frac{n + 1}{n} \right) $ but I feel like I am stuck on some simple algebra. Any hints for proving the sequence is nonincreasing? Or am I just wrong?

EDIT: Could this be done using absolute convergence? EDIT2: I am seeing some really great answers, but I am trying to prove this without calculus. (No derivatives or Taylor series expansions.)

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Did you notice that $\lim_{n\to\infty} n \ln \left( \frac{n + 1}{n} \right)=\lim_{n\to\infty} \ln \left( 1+ \frac{1}{n} \right)^n=\ln e=1$? – draks ... Jul 25 '12 at 8:19
hint: expand for example in Taylor series in $\frac 1n$. – Raymond Manzoni Jul 25 '12 at 8:21
Have you tried to consider the function $f \colon x \mapsto 1-x\log{\frac{x+1}{x}}$? It is differentiable, so you can study the sign of $f'$ in order to get information on the monotonicity... – Romeo Jul 25 '12 at 8:22
@draks - You should post that as an answer. In combination with the second sentence of the body of the question, this is a complete solution. – user22805 Jul 25 '12 at 8:25
@DavidWallace Huh? This leaves out the monotonicity. – Did Jul 25 '12 at 8:41
up vote 4 down vote accepted

Using the Taylor expansion of $\log(1+x)$ we see that $$ 1-n\log\Bigl(\frac{n+1}{n}\Bigr)=1-n\Bigl(\frac{1}{n}-\frac{1}{2\,n^2}+O(n^{-3})\Bigr)=\frac{1}{2\,n}+a_n,\quad a_n=O(n^{-2}). $$ Then $$ \sum_{n=1}^\infty(-1)^{n+1}\Bigl(1-n\log\Bigl(\frac{n+1}{n}\Bigr)\Bigr)=\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{2\,n}+\sum_{n=1}^\infty(-1)^{n+1}a_n. $$ The first series on the right hand side converges by Leibniz's test, and the second is absolutely convergent since $|(-1)^{n+1}a_n|\le C\,n^{-2}$ for some constant $C>0$.

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Look at $x \mapsto 1-x\log\left(\frac{x+1}{x}\right)$. Its second derivative is positive so its first derivative is increasing. It first derivative tends to $-\infty$ in when $x$ goes to zero and to $0$ when $x$ goes to $\infty$, thus it is negative, and the initial function is decreasing. Consider its restriction to $\mathbb{N}$ then.

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Notice that $$\lim_{n\to\infty} n \ln\left( \frac{n + 1}{n} \right)=\lim_{n\to\infty} \ln\left( 1+ \frac{1}{n} \right)^n=\ln e=1.$$

So $\lim_{n \to \infty} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) = 0$ as required and we are left with monotonicity:

$$ 1- (n + 1) \log \left( \frac{n + 2}{n + 1} \right) \leq 1- n \log \left( \frac{n + 1}{n} \right)\\ \log \left( \frac{n + 2}{n + 1} \right)^{n+1} \geq \log \left( \frac{n + 1}{n} \right)^n\\ \left( \frac{n + 2}{n + 1} \right)^{n+1} \geq \left( \frac{n + 1}{n} \right)^n\\ \left( 1+ \frac1{n+1} \right)^{n+1} \Biggr/ \left(1+ \frac1n \right)^n \geq 1\\ \left(\left( 1+ \frac1{n+1} \right) \Biggr/ \left(1+ \frac1n \right) \right)^n \geq \frac{n+1}{n+2}\\ $$ Now use $ \left(\left( 1+ \frac1{n+1} \right) \Biggr/ \left(1+ \frac1n \right) \right)^n \geq \left(\left( 1+ \frac1{n+1} \right) \Biggr/ \left(1+ \frac1n \right) \right)=\frac{n(n+2)}{(n+1)^2}$ to conclude: $$ \frac{n(n+2)}{(n+1)^2} \geq \frac{n+1}{n+2}\\ \frac{n(n+2)^2}{(n+1)^3}=\frac{n^3+4n^2+4n}{n^3+3n^2+3n+1} \geq 1\; ,\\ $$ for $n\geq 1$, which is not elegant, I know.

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Note that that's not enough for the alternating series to converge. The function needs to go to zero monotonically. It's easy to come up with an example of a function going to zero not monotonically that makes the alternating series divergent. – bartgol Jul 25 '12 at 8:46
But $(1 + \frac{1}{n})^{n}$ increases monotonically to $e,$ though that takes a little work and thought. So (the absolute values of) the terms in the quantity being summed do decrease monotonically, and it is OK in this case. – Geoff Robinson Jul 25 '12 at 9:03
@GeoffRobinson Would you mind having a look at my (not very elegant) proof for the monotonicity? – draks ... Jul 25 '12 at 14:17
Replace $+2n$ by $+4n$ at the very end. – Did Jul 25 '12 at 14:55
It looks correct, apart from the 2n-4n typo which did points out. A few more words might help though :) – Geoff Robinson Jul 25 '12 at 15:22

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