Is there a nice example of a convergent sequence $a_n \in \mathbb{C}$ such that
$$\sum\limits_{i=1}^{\infty} |a_i - a_{i+1}|$$
diverges? For example, can one find a Cauchy sequence $a_n$ such that $\frac{1}{n} = |a_n - a_{n+1}|$?
$a_n$ is Cauchy, but $\sum\limits_{i=1}^{\infty} |a_{i+1} - a_i|$ diverges
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real-analysis
sequences-and-series
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14$$a_n=\frac{(-1)^n}n$$ – 2017-01-22
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0@Did Your comment would be a nice answer. – 2017-01-22
1 Answers
1
One general approach would be to look for a convergent, but not absolutely convergent, series $\sum _{n=0} ^\infty x_n$ and then define $(a_i)_i$ recursively such that $a_{i+1} - a_i = x_i$, with $a_0$ arbitrary, which gives
$$a_{i+1} = a_0 + \sum _{n=0} ^i x_n .$$
$(a_i)_i$ converges to $a_0 + \sum _{n=0} ^\infty x_n$, therefore is clearly Cauchy, while $\sum _{i=0} ^\infty |a_i - a_{i+1}| = \sum _{i=0} ^\infty |x_i| = \infty$ by the assumption on $(x_n)_n$.
One particular such choice is $x_n = \frac {(-1)^n} n$, leading to $|a_n - a_{n+1}| = \frac 1 n$.