Question

Check the convergence$\sum _{k=2}^{\infty }\frac{1}{\sqrt[2]{e}}$

Short answer

Diverges.

Step-by-step solution

Step 1. Given

The given series is$\sum _{k=2}^{\infty }\frac{1}{\sqrt[2]{e}}$.

Step 2. Divergence test

According to the divergence test if the sequence {a_k} does not converge to zero then the series$\sum _{k=1}^{\infty }{a}_k$diverges .

Step 3. Checking the convergence 

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{sequence}{a}_k \\ {a}_k=\frac{1}{\sqrt[k]{e}} \\ =\frac{1}{e^{{\displaystyle \frac{1}{k}}}} \\ \mathrm{The}\mathrm{value}\mathrm{of}\underset{k\to \infty }{\lim }{a}_{k}is \\ \underset{k\to \infty }{\lim }{a}_k=\underset{k\to \infty }{\lim }\frac{1}{e^{{\displaystyle \frac{1}{k}}}} \\ =1 \\ \mathrm{Since},\underset{k\to \infty }{\lim }{a}_k\cong 0 \\ \mathrm{Here}{\mathrm{a}}_{\mathrm{k}}\mathrm{does}\mathrm{not}\mathrm{converge}\mathrm{to}0. \\ \mathrm{Hence},\mathrm{the}\mathrm{series}\mathrm{diverges}. \end{gathered}$$