Does the series ${a_n}$ = $\sum _1^n \frac{1}{n^{1+\alpha}}$ converge for all $\alpha$ > 0?
Asked
Active
Viewed 69 times
0
-
Are you wanting a sequence $a_n$ or a series $\sum a_n$? – Clayton Apr 19 '15 at 05:06
-
I think he meant partial sum $a_n$ – Jesse P Francis Apr 19 '15 at 05:06
-
I meant the partial sum – Sriram Natarajan Apr 19 '15 at 05:13
2 Answers
1
Yes. You can use the Integral Test.
Berrick Caleb Fillmore
- 5,269
-
-
1I’ve added a link to the Wikipedia article on the topic. – Berrick Caleb Fillmore Apr 19 '15 at 05:08
1
The series $\sum _{n=1}^{\infty} \frac{1}{n^{k}}$ converges for all $k>1$. It is divergent if $k \le 1$. It is a generalization of the Harmonic Series.
user5826
- 11,982