Question

For a convergent series satisfying the conditions of the integral test, why is every remainder $R_n$positive? How can $R_n$be used along with the term $S_n$ from the sequence of partial sums to understand the quality of the approximation $S_n$?

Short answer

The remainder is positive because the function is positive.

Then

$S_n\le \sum _{k=1}^{\infty }a\left(k\right)\le S_n+B_{n}where{B}_n={\int }_n^{\infty }a\left(x\right)dx$

If the remainder is small, the quality of approximation is good.

Step-by-step solution

Step 1. Given Information.

The convergent series satisfying the conditions of the integral test.

Step 2. Approximating the Remainder for a Series That Converges by the Integral Test.

If a function a is continuous, positive, and decreasing, and if the improper integral ${\int }_1^{\infty }a\left(x\right)dx$ converges, then the nth remainder, $R_n$, for the series $\underset{k=1}{\overset{\infty }{\sum a\left(k\right)}}$is bounded by,

$0\le R_n\le {\int }_n^{\infty }a\left(x\right)dx$.

The remainder is positive because the function is positive.

Then

$S_n\le \sum _{k=1}^{\infty }a\left(k\right)\le S_n+B_{n}where{B}_n={\int }_n^{\infty }a\left(x\right)dx$

If the remainder is small, the quality of approximation is good.