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JG89
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Homework Statement
I was reading a question in a textbook that said, "Prove that [tex] \sum_{v=1}^{\infty} \frac{a_v^n}{v} = ln(n) [/tex] where [tex] a_v^n = 1 [/tex] if n is not a factor of v, and [tex] a_v^n = -(n-1) [/tex] if n is a factor of v.
Homework Equations
The Attempt at a Solution
The author started his proof as follows:
Take the sum from v = 1 to v = mn:
[tex] \sum_{v=1}^{v=mn} \frac{a_v^n}{v} = \sum_{v \ne mn} \frac{1}{v} - \sum_{v=kn} \frac{n-1}{v} [/tex].
Didn't the author just rearrange the terms, even though the series cannot be absolutely convergent (this is easy to prove), and so he could have just changed the sum of the series?
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