Originally Posted by

**utopiaNow** Here are the 2 series where their nth terms are shown:

$\displaystyle

c_n = (-1)^n \frac{(n+1)^n}{n^n}

$

$\displaystyle

d_n = \frac{(n+1)^n}{n^{n + 1}}

$

Attempted solution:

So I know about all the tests: monotone convergence criterion, comparison test, root test, ratio test, and alternating series test.

For $\displaystyle c_n $ I tried to use the alternating series test however it's neither nonincreasing nor clear if $\displaystyle \displaystyle\lim_{n\to\infty}c_n = 0$.

For $\displaystyle d_n $ my hunch is to say that since $\displaystyle n \geq 1 \Rightarrow 1^n \leq \frac{(n+1)^n}{n^{n + 1}}$. Therefore by the comparison test, since $\displaystyle 1^n$ diverges so does $\displaystyle d_n$.