Absolutely Convergent

College/University

Definition

Also known as absolute convergence, this refers to an infinite series that converges when its terms are changed to their absolute values. This is done by changing any subtraction signs in the series to additions, then checking if the new series converges.

Worked examples

\(\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}\) is absolutely convergent because \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges.

Replace each term with its absolute value; the resulting positive series converges, so the original is absolutely convergent.

\(\sum_{n=1}^{\infty} \frac{\sin(n)}{n^3}\) is absolutely convergent because \(\sum_{n=1}^{\infty} \frac{|\sin(n)|}{n^3} \le \sum_{n=1}^{\infty} \frac{1}{n^3}\) converges.

Bound the absolute values and use the comparison test to show the absolute-value series converges.

Common mistakes

\(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) is absolutely convergent because it converges by alternating series test → It is conditionally convergent; \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges Absolute convergence requires the series of absolute values to converge, not just the original series.
If \(\sum a_n\) converges, then \(\sum |a_n|\) converges → Convergence does not imply absolute convergence The harmonic series with alternating signs converges, but its absolute values diverge.
Only use absolute convergence tests for series with negative terms → Absolute convergence applies to any series and guarantees standard convergence If a series is absolutely convergent, you can rearrange its terms without changing the sum.

Where you'll use it next

Absolute convergence is essential in advanced calculus for power series, Fourier series, and proving that rearranging terms does not change the sum. It appears in analysis, differential equations, and complex variables.

Found in 1 StudyPug lesson

Mastering Absolute and Conditional Convergence in Series

Calculus 2

Dive deep into absolute and conditional convergence, crucial concepts for analyzing infinite series. Enhance your calculus skills and tackle advanced mathematical problems with confidence.

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