Show that the series is convergent. How many terms of the...

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - \frac {1}{3})^n} {n} (|error| < 0.0005) $

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Key Concepts

Absolute Convergence

A series is said to be absolutely convergent if the series of absolute values converges. Absolute convergence is a stronger condition than conditional convergence, and if a series is absolutely convergent, then it converges regardless of the signs of its terms. This concept is often used to reinforce convergence conclusions obtained from the alternating series test.

Error Estimation for Alternating Series

When using the alternating series test, there is an associated error bound for approximating the sum by a finite partial sum. Specifically, the error in truncating the series is less than or equal to the absolute value of the first omitted term. This property is crucial for determining how many terms are needed to achieve a desired accuracy.

Alternating Series Test

This test is used for series in which the signs of the terms alternate. It states that if the absolute values of the terms are monotonically decreasing and tend to zero, then the series converges. This condition helps in establishing convergence without needing to sum the series explicitly.

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