# series tests

• Jul 5th 2007, 05:00 PM
viet
series tests
Quote:

Select the FIRST correct reason on the list why the given series converges.

a) Geometric series.
b) Comparison with a convergent p series.
c) Integral test.
d) Ratio test.
e) Alternating series test.

1) $\displaystyle \sum_{n=1}^\infty \frac{1}{n\left(ln\left(n\right)\right)^2}$

2) $\displaystyle \sum_{n=1}^\infty \frac{\sin^2 (4 n)}{n^2}$

3) $\displaystyle \sum_{n=1}^\infty\frac{n^2+\sqrt{n}}{n^4 -3}$
im struggling a bit on series section, how do you know what the best test for each problem?
• Jul 5th 2007, 05:18 PM
ThePerfectHacker
1)Use the integral test with $\displaystyle \int_2^{\infty} \frac{dx}{x \ln^2 x }$ (I think your index such start with 2).

2)This is a compasion test $\displaystyle \left| \frac{\sin ^2 (4n)}{n^2 } \right| \leq \frac{1}{n^2}$ and $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty$.

3)Another comparasion test $\displaystyle \left| \frac{n^2 + \sqrt{n}}{n^4 - 3} \right| \leq \frac{n^2+n^2}{n^4 - \frac{1}{2}n^4} = \frac{4n^2}{n^4} = \frac{2}{n^2}$.

This is Mine 63:):)th Post!!!
• Jul 5th 2007, 06:02 PM
viet
thank you ThePerfectHacker and congratulation