Step by Step Solutions

[Steps Shown] Use the following steps to show that ∑_n=1^∞ ((-1)^n-1)/(n)=ln 2 Let h_n and s_n be the partial sums of the harmonic and alternating

Question: Use the following steps to show that

\[\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\ln 2\]

Let \(h_{n}\) and \(s_{n}\) be the partial sums of the harmonic and alternating harmonic series.

  1. Show that \(s_{2 n}=h_{2 n}-h_{n}\).
  2. From Exercise 40 in Section 12.3 we have

\[h_{n}-\ln n \rightarrow \gamma \quad \text { as } n \rightarrow \infty\]

and therefore

\[h_{2 n}-\ln (2 n) \rightarrow \gamma \quad \text { as } n \rightarrow \infty\]

Use these facts together with part (a) to show that \(s_{2 n} \rightarrow \ln 2\) as \(n \rightarrow \infty\)

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