Question

Simplify

k^k/(k+1)^(k+1)

Answer 1

meaning \[(k)^{k}/(k+1)^{k+1}\]

Answer 2

( k +1) ^ (k +1) = ( k +1) ^ k * ( k + 1)

Answer 3

yep i agree with that, thats the bottom only
what about the top

Answer 4

thats my issue, im trying to simplify an infinite series using the ratio test and cant figure out how to simplify this part to make stuff cancel

Answer 5

It's not possible to simplify, since no divisor of \(k\) will divide \(k+1\).

Answer 6

(Or the other way around)

Answer 7

But... I will say that using the ratio test, the series converges absolutely, since \(\frac{k^k}{(k+1)^{k+1}}<\frac{k^k}{k^{k+1}}\) whose limit is actually zero. But, I don't know if you meant it the other way around since it is:
\[
\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|
\]