Q. 16 - Calculus

Question

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

Step-by-Step Solution

Answer

Hence proved.

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.