Q. 15 - Calculus

Question

Fill in the blanks: Let f be a function with domain $[1,\infty )$. If the function $\left|f\right|$ is ______ and ______ , and if ______ converges, then the series _______ converges absolutely.

Step-by-Step Solution

Answer

On completing the fill in the blanks, we get, "Let f be a function with domain $[1,\infty )$. If the function $\left|f\right|$ is monotonically and decreasing , and if $\underset{x\to \infty }{\lim }f\left(x\right)=0$ converges, then the series ${\int }_{x=1}^{\infty }f\left(x\right)$ converges absolutely. "

Step 1. Given information.

Consider the given question,

Domain is $[1,\infty )$.

Step 2. Fill in the blanks.

If the function $\left|f\right|$ is monotonically decreasing and $\underset{x\to \infty }{\lim }f\left(x\right)=0$ and the integral ${\int }_{x=1}^{\infty }f\left(x\right) dx$ converges, then the series ${\int }_{x=1}^{\infty }f\left(x\right)$ converges absolutely by integral test.

Therefore, the first blank is completed by monotonically and decreasing, second blanks by $\underset{x\to \infty }{\lim }f\left(x\right)=0$ and third is ${\int }_{x=1}^{\infty }f\left(x\right)$.