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Consider the power series $\sum_{n\ge1} a_n z^n$ where $a_n =$ number of divisors of $n^{50}$. then the radius of convergence of $\sum_{n\ge1} a_n z^n$ is

(1) 1

(2) 50

(3) $\frac 1 {50}$

(4) 0

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    So, what do you know about the number of divisors of $n^{50}$?2012-11-16
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    @Gerry, I see your point.2012-11-16
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    @Gerry if n is prime then the number of divisors of n^50 is 51, but when n is composite..?2012-11-16
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    You don't need an exact answer, just a bound good enough to tell you when the series converges.2012-11-16

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Hint: what is the radius of convergence of $\sum_{n\ge1}z^n$? of $\sum_{n\ge1}n^{50}z^n$?

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    @ Gerry 1 is the radius of convergence of ∑n≥1zn2012-12-15
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    Yes, the radius of convergence of $\sum_{n\ge1}z^n$ is $1$. Can you work out the radius of convergence of the other sum I mention? maybe using the ratio test?2012-12-15