Section 10.13

Estimating the Value of a Series

We know it converges, but to what? And how close are we?

Introduction

Most series cannot be summed exactly (unlike Geometric or Telescoping series). Instead, we approximate the sum using the -th partial sum .

The error (remainder) is . Our goal is to find an upper bound for .

is continuous, positive, and decreasing, then:

This gives us a range for the error. We can also average the bounds to get a better estimate.

For alternating series , the error is simply less than the next term.

The true sum (dashed line) is always within (Red Bar) of the partial sum.

Estimate with error .

We need .

.
Check powers: , .
So .

Sum first 5 terms:
.

Estimate error for with .

Error .
.

Error is at most 0.1.