The Integral Test

1. Determine if the following p-series converges or diverges.

(a) \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^4}\)	(d) \(\displaystyle 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots\)
(b) \(\displaystyle \sum_{n=1}^{\infty} \frac{2}{\sqrt[3]{n^2}}\)	(e) \(\displaystyle 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \frac{1}{\sqrt{5}} + \cdots\)
(c) \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{3}}}\)	(f) \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt[\pi]{n^4}}\)

(a) \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+n^2}\) View Video Solution	(b) \(\displaystyle \sum_{n=1}^{\infty} \frac{\sqrt[3]{n}+2}{n^2}\) View Solution 1 View Solution 2
(c) \(\displaystyle \sum_{n=1}^{\infty} \frac{n}{n^2+4}\) View Video Solution	(d) \(\displaystyle \sum_{n=2}^{\infty} \frac{\ln n}{n}\) View Video Solution
(e) \(\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\ln n}\) View Video Solution	(f) \(\displaystyle \sum_{n=1}^{\infty} n^2 e^{-n^3}\) View Video Solution
(g) \(\displaystyle \sum_{n=2}^{\infty} \frac{e^{1/n}}{n^2}\) View Video Solution	(h) \(\displaystyle \sum_{n=3}^{\infty} \frac{n^2}{e^n}\) View Video Solution