Question

Check the convergence$\sum _{k=1}^{\infty }{(-1)}^k\ln k$

Short answer

The series diverges.

Step-by-step solution

Step 1. Given

The given series$\sum _{k=1}^{\infty }{(-1)}^k\ln k$

Step 2. Checking of convergence

The general term of the series $\sum _{k=1}^{\infty }{(-1)}^k\ln k$is $a_k=\ln k$

localid="1649739242985" $$\begin{gathered} \mathrm{The}\mathrm{value}\mathrm{of}\underset{k\to \infty }{\lim }{a}_{k}is: \\ \underset{k\to \infty }{\lim }{a}_k=\underset{k\to \infty }{\lim }\ln k \\ =\infty \\ \mathrm{Since}\underset{k\to \infty }{\lim }{a}_k\varnothing 0 \\ \mathrm{Hence}\mathrm{the}\mathrm{series}\underset{k\to \infty }{\lim }\ln k\mathrm{diverges}. \end{gathered}$$