Question

Test the convergence of the series using root test

Answer 1

\[\sum_{n=1}^{\infty}(\frac{ 2n+3 }{ 3n+2 })^{n}\]

Answer 2

\[
\lim_{n\to \infty } \,
\left(\left(\frac{2 n+3}{3
n+2}\right)^n\right)^{1/n}=\lim_{
n\to \infty } \, \frac{2 n+3}{3
n+2}=\frac{2}{3}<1

\]
It converges.