Convergence of a Particular Sequence

Question

I was supposed to use the convergence tests to determine whether the following series converges, converges absolutely, or diverges.

Almost every test I used was inconclusive, the series is : $$\sum_{k=1}^{\infty} (-1)^k \frac{k\ln(k)}{(k+1)^3}.$$

Answer 1

Consider,

$$f(x)=\frac{x \ln x}{(x+1)^3}$$

Then,

$$f'(x)=\frac{x-2x\ln x+\ln x+1}{(x+1)^4}$$

One can show that for $x>3$ that we must have $x-2x\ln x+\ln x+1<0$ by considering the monotonically decreasing function $g(x)=x-2x\ln x+\ln x+1$ defined on $(1,\infty)$. I leave it to you to fill in the gaps. Thus we have for $x>3$ that $f'(x)<0$. Hence it follows that,

$$\frac{n \ln n}{(n+1)^3}$$

Is eventually decreasing, and your series converges by the alternating series test.

Answer 2

You have that $$ \frac{n\ln n}{(n+1)^3} \sim_{n\to \infty} \frac{\ln n}{n^2} $$ so that, since the (positive) series $\sum_n \frac{\ln n}{n^2}$ converges (e.g., by comparison with a $p$-series $\sum_n \frac{1}{n^{3/2}}$) by theorems of comparison the series $\sum_n (-1)^n \frac{n\ln n}{(n+1)^3}$ converges absolutely.