# Proving convergence of 1/log(n)

by hivesaeed4
Tags: 1 or logn, convergence, proving
 P: 212 Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)
 P: 590 0 < log(n) < n for n>3
P: 606
 Quote by hivesaeed4 Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)

First way, the integral test: $\int_2^\inf \frac{1}{x\log^px} dx=\frac{\log^{1-p}(x)}{1-p}|_2^\inf \rightarrow \frac{log^{1-p}(2)}{p-1}$ .

Second way, Cauchy's Condensation test: taking $n=2^k$ , the series's general term is $\frac{1}{2^kk^p\log^p(2)}$ , so multiplying this by $2^k$ we get $\frac{1}{k^p\log^p(2)}$ , which is a multiple of the series of $\frac{1}{k^p}$ , which we know converges for $p>1$ .

DonAntonio

P: 212

## Proving convergence of 1/log(n)

 Quote by DonAntonio First way, the integral test: $\int_2^\inf \frac{1}{x\log^px} dx=\frac{\log^{1-p}(x)}{1-p}|_2^\inf \rightarrow \frac{log^{1-p}(2)}{p-1}$ . Second way, Cauchy's Condensation test: taking $n=2^k$ , the series's general term is $\frac{1}{2^kk^p\log^p(2)}$ , so multiplying this by $2^k$ we get $\frac{1}{k^p\log^p(2)}$ , which is a multiple of the series of $\frac{1}{k^p}$ , which we know converges for $p>1$ . DonAntonio
Thing is I've got to prove determine it's convergence from 1 to infinity.
 P: 212 Just to clarify, the previous message was supposed to be: Thing is I've got to determine it's convergence from 1 to infinity.
 Mentor P: 4,258 If the sum starts at 1, you have a problem because log(1)=0
 P: 590 The sum from 1 to any fixed number will always converge (unless you divide by zero). Therefore, if the sum from any fixed number to infinity converges, then the sum from 1 to infinity converges. You can choose any fixed number you like. I chose 3 because 0=3. You can choose 11 if your log is base 10 rather than natural, or because, you know, usual amps only go up to 10, but yours goes all the way up to 11. The sum from 3 to infinity of your series is smaller than the sum from 3 to infinity over 1/n^(p+1), which is smaller than 1/n^2 which we know converges.

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