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Question

# Show that the series $\sum_{j=1}^{\infty}\left(1-2^{-j}\right) / j$ diverges.

Solution

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Step 1
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Firstly, test the series

$\sum_{j=1}^{\infty}\frac{(1-2^{-j})}{j}$

to the neccesary convergence condition:

\begin{align*} \lim_{n\rightarrow\infty} \frac{(1-2^{-n})}{n}&=\lim_{n\rightarrow\infty}\left(\frac{1}{n}-\frac{1}{n\cdot2^n} \right) \,,\\ &\overset{(\ast)}{=}\lim_{n\rightarrow\infty}\left(\frac{1}{n}\right)-\lim_{n\rightarrow\infty}\left(\frac{1}{j\cdot2^j}\right)\,,\\ &=0-0\,,\\ &=\underline{0}\,.\quad\color{#c34632}\checkmark \end{align*}

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