Analyzing Convergence of 1/n(ln n)^p and 1/((n(ln n)(ln (ln n)))^p

[jkh4]

Hi, I just wondering if someone can help me on this question.

1) 1/(n(ln n ) ^p)

2) 1/((n(ln n)(ln (ln n))))^p

Both are b = infinity and a = 1 for the integral signs.

The question is for what value of p>0 does the series converges and for what value does it diverges.

Thank you !

 

[StatusX]

There was a similar question in the HW section recently. Use the integral test.

 

[quasar987]

Do you know about the criterion $\sum a_n$ converges iff $\sum 2^n a_{2^n}$ converges? It is usefull for series involving ln(n) because ln(2^n)=nln(2) !

Then, for the values of p, use the ratio test (a_{n+1}/a_n).

 

[StatusX]

Do you know about the criterion $\sum a_n$ converges iff $\sum 2^n a_{2^n}$ converges?

It's important to keep in mind this only works when the series is monotonically decreasing (which this one is). Otherwise, just take the terms to be all 1's except for 0's when n is a power of 2.