Estimating the Sum of a Series

Summary

When a series converges, partial sums $S_{N}$ approximate the total sum $S$ . Remainder bounds quantify the error $| S - S_{N} |$ using integral tails, alternating-series estimates, or exact remainders for geometric series.

Prerequisites

Series Sums by Partial Sums, Geometric Series, Alternating Series, Integral Test

Formulas

Geometric series (exact)

For $| r | < 1$ ,

\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}, R_{N} = \sum_{n = N + 1}^{\infty} a r^{n} = \frac{a r^{N + 1}}{1 - r} .

Alternating series remainder

If $\sum (- 1)^{n + 1} b_{n}$ satisfies the alternating series test with $b_{n} ↓ 0$ , then

| S - S_{N} | \leq b_{N + 1} .

Integral remainder (decreasing positive $f$ )

If $a_{n} = f (n)$ with $f$ positive, continuous, and eventually decreasing,

\int_{N + 1}^{\infty} f (x) d x \leq R_{N} \leq \int_{N}^{\infty} f (x) d x

(under standard integral-test hypotheses).

Worked Example

For $\sum_{n = 0}^{\infty} (1 / 2)^{n}$ , the exact sum is $2$ . After three terms $S_{2} = 1 + 1 / 2 + 1 / 4 = 1.75$ , the remainder is

R_{2} = \frac{(1 / 2)^{3}}{1 - 1 / 2} = \frac{1}{4} = 0.25,

so $S = 1.75 + 0.25 = 2$ exactly—not a hand-wavy “about $1.8$ .”

For the alternating harmonic series, $| S - S_{5} | \leq 1 / 6$ .

Common Mistakes

Guessing remainders without a theorem (“looks like $1.8$ ”).
Broken MathJax such as $\ln$ n$ instead of $\ln n$ .
Using alternating bounds on non-alternating series.

Connections

Alternating Series, Integral Test, Taylor Series (Lagrange remainder)

References

Remainder estimates appear with the integral and alternating tests in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Sections 5.3–5.5, https://openstax.org/details/books/calculus-volume-2 ↩︎