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.

Short 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-by-step solution

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${\int }_{x=1}^{\infty }f\left(x\right)$.