Remainder estimate

Definition

A remainder estimate provides a bound on the error when approximating an infinite series by a partial sum. It helps determine how close the partial sum is to the actual value of the series.

5 Must Know Facts For Your Next Test

1. The remainder estimate for an alternating series can be found using the next term in the series.
2. For a convergent series, if $S$ is the sum and $S_n$ is the nth partial sum, then $R_n = S - S_n$ represents the remainder after $n$ terms.
3. In Integral Test, remainder estimates use integrals to bound the error.
4. The Remainder Estimate Theorem states that if $\sum a_n$ converges by Integral Test, then $R_n \leq \int_{n}^{\infty} f(x) \, dx$.
5. Understanding remainder estimates is crucial for practical applications of series approximations.

Review Questions

• How do you express the remainder after summing up to the nth term?
• What does the Remainder Estimate Theorem state in relation to integral bounds?
• How can you find a remainder estimate for an alternating series?

Related terms

Partial Sum: The sum of a finite number of initial terms of a sequence.

Alternating Series: A series whose terms alternate in sign.

Integral Test: \text{A method used to determine whether an infinite series converges or diverges based on evaluating an improper integral.}