Use the Comparison Test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { n } { 2 n + 5 } \right) ^ { n }$$
A) converges
B) diverges

Question

Quizplus · Accepted Answer

We use the Comparison Test by comparing the given series with a geometric series. Notice that for $n\geq 1$, $\left( \frac { n } { 2 n + 5 } \right) ^ { n }\leq \left( \frac {1} {3} \right) ^n$, since 
$$\left( \frac { n } { 2 n + 5 } \right) ^ { n } = \frac{1}{\left(1+ \frac{5}{2n}\right)^n}\leq \frac{1}{\left(1+ \frac{5}{2}\right)^n}=\frac{1}{\left(\frac{7}{2}\right)^n}=\left(\frac{2}{7}\right)^n\cdot\frac{1}{n!}$$
which follows from the famous inequality $a_n=(1+x/n)^n<1/x^n\cdot n!$ for all $n$ and $x>0.$ (We take $x=5/2.$)
But $\sum _ { n = 1 } ^ { \infty }\left( \frac {1} {3} \right) ^n$ converges since it is a geometric series with ratio less than 1. Hence by the Comparison Test, the given series also converges.

Use the Comparison Test to Determine If the Series Converges ∑n=1∞(n2n+5)n\sum _ { n = 1 } ^ { \infty } \left( \frac { n } { 2 n + 5 } \right) ^ { n }n=1∑∞​(2n+5n​)n

Use the Comparison Test to Determine If the Series Converges $\sum _ { n = 1 } ^ { \infty } \left( \frac { n } { 2 n + 5 } \right) ^ { n }$