Q. 13 - Calculus

Question

Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series $\sum _{k=1}^{\infty }{a}_k$ for convergence.

Step-by-Step Solution

Answer

The function a(x) has to be continuous because if the function is not continuous, the improper integral ${\int }_1^{\infty }a\left(x\right)dx$ may not be defined.

So if the integral is not defined, the convergence or divergence cannot be examined.

Step 1. Given Information.

The function: a(x)

The series:

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

Step 2. The integral test.

If $f\left(x\right):[1,\infty )\to \mathrm{ℝ}$ is continuous, eventually positive and decreasing on $[1,\infty )$, and $f_k$is the sequence defined by $f_k=\left\{f\right(k\left)\right\}$ for every $k\in {\mathrm{\mathbb{Z}}}^{+}$ , then $\sum _{k=1}^{\infty }{f}_k and {\int }_1^{\infty }f\left(x\right)dx$ either both converge or diverge.

Step 3. Why it has to be continuous.

The function a(x) has to be continuous because if the function is not continuous, the improper integral  may not be defined.

So if the integral is not defined, the convergence or divergence cannot be examined.