Question

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

$\sum _{k=1}^{\mathrm{\infty }} \frac{1}{k^k}$

Short answer

Ans: The divergence test fails divergent because,$\sum _{k=1}^{\mathrm{\infty }} \frac{1}{k^k}=0$

Step-by-step solution

Step 1. Given information.

given,

$\sum _{k=1}^{\mathrm{\infty }} \frac{1}{k^k}$

Step 2. The objective is to use the divergence test to analyze the given series.

The divergence test states that if the sequence $\left\{a_k\right\}$does not converge to zero, then the series localid="1649063033376" $\sum _{k=1}^{\mathrm{\infty }} a_k$diverges.

The value of the sequence localid="1649063415009" $\left\{a_k\right\}=\left\{\frac{1}{k^k}\right\}$is:

$\begin{array}{r}\underset{k\to \mathrm{\infty }}{lim} a_k=\underset{k\to \mathrm{\infty }}{lim} \left(\frac{1}{k^k}\right)=0\\ \end{array}$

The divergence test fails divergent because,$\underset{k\to \mathrm{\infty }}{lim} \left(\frac{1}{k^k}\right)=0$