Question 8.5.1: Testing for Absolute Convergence Determine whether ∑^∞k=1 (−...
Testing for Absolute Convergence
Determine whether \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2^k} is absolutely convergent.
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It is easy to show that this alternating series converges to approximately
0.35. (See Figure 8.34.) To determine absolute convergence, we need to check whether or not the series of absolute values is convergent. We have
which you should recognize as a convergent geometric series \left(|r|=\frac{1}{2}<1\right). This says that the original series \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2^k} converges absolutely.

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