Absolute Convergence

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Definition

Also known as absolutely convergent, 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 (a p-series with p=2) converges.

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

Bound the absolute values and use the comparison test; the dominating series converges.

Common mistakes

\(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) is absolutely convergent because it converges by the alternating series test. → \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) is conditionally convergent; \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges. Absolute convergence requires the series of absolute values to converge; this harmonic series diverges.
If \(\sum a_n\) converges, then \(\sum |a_n|\) converges. → If \(\sum |a_n|\) converges, then \(\sum a_n\) converges. Absolute convergence implies convergence, but convergence does not imply absolute convergence.
To test absolute convergence, use the alternating series test on \(\sum |a_n|\). → Use ratio test, root test, or comparison tests on \(\sum |a_n|\). The alternating series test applies only to alternating series, not to the positive-term series of absolute values.

Where you'll use it next

Absolute convergence is essential in advanced calculus for rearranging series terms, proving convergence of power series and Fourier series, and in real analysis when studying function spaces and uniform convergence.

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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