Q. 8 - Calculus

Question

Explain how you could adapt the integral test to analyze a series $\sum _{k=1}^{\infty }f\left(k\right)$ in which the function $f:[1,\infty )\to \mathrm{ℝ}$ is continuous, negative, and increasing.

Step-by-Step Solution

Answer

By the integral test, the series $\sum _{k=1}^{\infty }f\left(k\right)$ is divergent.

Step 1. Given Information.

The series:

$\sum _{k=1}^{\infty }f\left(k\right)$

Step 2. Integral test.

By the integral test, the function will both converge or diverge. 

$$\begin{gathered} {\int }_1^{\infty }-f\left(x\right)dx=\underset{k\to \infty }{lim}{\int }_1^k-\alpha .dx \\ =\underset{k\to \infty }{lim}{[-\alpha x]}_1^k \\ =\underset{k\to \infty }{lim}[-\alpha k+\alpha ] \\ =0 \end{gathered}$$

Step 3. Convergent or divergent.

By the integral test, the improper integral ${\int }_1^{\infty }f\left(x\right)dx$ is divergent.

So the series $\sum _{k=1}^{\infty }f\left(k\right)$ is divergent.