Question

Explain why the series $\sum _{k=1}^{\infty }\frac{1}{k^3}$ converges. Which convergence tests could be used to prove this?

Short answer

Hence proved.

Step-by-step solution

Step 1. Given information.

We are given$\sum _{k=1}^{\infty }\frac{1}{k^3}$.

Step 2. Explanation.

Now, According to Divergence and convergence test,

$$\begin{gathered} \sum _{k=1}^{\infty }\frac{1}{k^3}\text{is of the form}\sum _{k=1}^{\infty }\frac{1}{k^p}\text{, where}p=3\text{.} \\ p>1\text{, therefore, the series}\sum _{k=1}^{\infty }\frac{1}{k^3}\text{converges.} \end{gathered}$$

The test which could be used to test the convergence of the series are Ratio test, Root Test.