Q. 24 - Calculus

Question

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

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

Step-by-Step Solution

Answer

Ans:  The series  $\sum _{k=1}^{\mathrm{\infty }} \frac{k}{e^k}$is convergent.

Step 1. Given information.

given,

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

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series.

Consider function $f\left(x\right)=x{e}^{-x}$.

The first derivative of the function $f\left(x\right)=x{e}^{-x}$ is:

$\begin{array}{r}{f}^{\mathrm{\prime }}\left(x\right)=e^{-x}-x{e}^{-x}\\ =(1-x)e^{-x}\end{array}$

The derivative is negative for all $x>1$. Therefore the function is decreasing.

The function $f\left(x\right)=x{e}^{-x}$ is continuous, decreasing, with positive terms. Therefore, all the conditions of the integral test are fulfilled. So, the integral test is applicable.

Step 3. To evaluate the integral, integrate by parts.

Therefore,

         $\begin{array}{r}{\int }_{x=1}^{\mathrm{\infty }} x{e}^{-x}dx=\underset{k\to \mathrm{\infty }}{lim} {\int }_{x=1}^k x{e}^{-x}dx \\ =\underset{k\to \mathrm{\infty }}{lim} {\left[-x{e}^{-x}-\left(e^{-x}\right)\right]}_{x=1}^k \text{ (Integrate) }\\ =\underset{k\to \mathrm{\infty }}{lim} {\left[-x{e}^{-x}-e^{-x}\right]}_{x=1}^k \\ =\underset{k\to \mathrm{\infty }}{lim} \left[-k{e}^{-k}-e^{-k}+2e^{-1}\right] \text{ (Substitution) }\\ =2e^{-1} \text{ (Take limit) }\end{array}$

Step 4. Thus, the value of the integral is ∫ x = 1 ∞   x e − x d x = 2 e .

The integral converges. Therefore, the series $\sum _{k=1}^{\mathrm{\infty }} \frac{k}{e^k}$ is convergent.

Hence, by integral test, the series $\sum _{k=1}^{\mathrm{\infty }} \frac{k}{e^k}$ is convergent.