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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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    @DavidWallace Would you mind having a look at my (not very elegant) proof for the monotonicity?2012-07-25

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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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    It looks correct, apart from the 2n-4n typo which did points out. A few more words might help though :)2012-07-25
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Recall that for all $x>0$ $ 1 - \frac1x \leq \log x \leq x-1 \tag{1} $

If $x=\frac{n+1}{n}$, then $x>1$ and $n=\frac{1}{x-1}$. Multiplying $({1})$ by $-n=\frac{1}{1-x}<0$ and then adding $1$ we have $ 1-n+\frac{n}{x}\geq1-n\log x \geq 1-nx+n $ or $ 1-n+\frac{n^2}{n+1}\geq1-n\log \frac{n+1}{n} \geq 1-n\frac{n+1}{n}+n=0 $ A more explicit estimate would be $ 1-\frac{1}{n+1}\geq1-n\log \frac{n+1}{n} \geq 0 \tag{2} $

By the criterion of convergence of alternating series we have that the series $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - \frac{1}{n+1}\right) $ converges. By the criterion of comparison for alternating series, it follows from $(2)$ that the series $ \sum_{n = 1}^{\infty} (-1)^{n+1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges.