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 {( - 1)^{n-1}}{n^2 2^n} (|error| < 0.0005) $

Key Concepts

Alternating Series Test

The alternating series test is a method used to determine the convergence of series whose terms alternate in sign. It requires that the absolute value of the terms decreases monotonically to zero. If both conditions are met, the series converges. This test is essential when examining series with alternating negative and positive terms.

Absolute Convergence

Absolute convergence refers to a situation where the series of absolute values converges. If a series is absolutely convergent, it guarantees the convergence of the original series regardless of the ordering of its terms. This concept is important because many alternating series are also absolutely convergent, providing a stronger form of convergence.

Error Estimation in Alternating Series

The error estimation in alternating series is based on the alternating series estimation theorem, which states that the remainder (error) after summing n terms is less than or equal to the absolute value of the (n+1)th term. This property is particularly useful for determining how many terms are needed to achieve a specified level of accuracy in the sum of the series.

Partial Sums and Convergence Accuracy

The concept of partial sums involves adding a finite number of terms from an infinite series to approximate its total sum. Understanding how the remainder (difference between the partial sum and the actual sum) behaves allows one to determine the number of terms required to achieve a desired accuracy. This is crucial when applying error bounds to ensure that the approximation meets a specified tolerance.

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