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Question

# Show how to rearrange the terms of the series from the specified exercise to form a divergent series.$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1+n}{n^2}$

Solution

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From the other exercise, we found that it conditionally converges (converges but not absolutely).

\begin{align*} \sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{n+1}{n^2} &= \dfrac{ 2}{1 } - \dfrac{ 3}{4 } + \dfrac{ 4}{9 } - \dfrac{ 5}{16 } + \dfrac{ 6}{25 } - \dfrac{ 7}{36 } + \dfrac{ 8}{ 49} \end{align*}

The positive terms diverge to $\infty$ while the negative terms diverge to $-\infty$.

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