Answer to Question #291363 in Real Analysis for Dhruv bartwal

Question

Examine the following series for convergence

∞Σn=1 [(n-1/(2n+3)]^n

Answer 1

Solution:

Let $a_n=(\dfrac{n-1}{2n+3})^n$

Now, consider $L=\lim_{n\rightarrow \infty} {a_n}^{1/n}$

$\Rightarrow L=\lim_{n\rightarrow \infty} [{(\dfrac{n-1}{2n+3})^n}]^{1/n} \\ \Rightarrow L=\lim_{n\rightarrow \infty} (\dfrac{n-1}{2n+3}) \\ \Rightarrow L=\lim_{n\rightarrow \infty} (\dfrac{1-\frac1n}{2+\frac3n}) \\ \Rightarrow L= \dfrac12$

Since, $L<1$, so by root test, $\Sigma a_n$ converges.