Series: The Integral Test    Lecture video

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

(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) \(\displaystyle{\sum_{n=1}^{\infty}\frac{1}{\sqrt[\pi]{n^4}}}\)

2. Use ''The Integral Test'' to determine if each of the following series converges or diverges. (visit ww2.coastal.edu/rdahal/math161/all_tests/ for alternative solutions)

(a) \(\displaystyle{ \sum_{n=1}^{\infty} \frac{1}{1+n^2}}\)   solution

(b) \(\displaystyle{\sum_{n=1}^{\infty} \frac{\sqrt[3]{n}+2}{n^2}}\)   solution 1   solution 2

(c) \( \displaystyle {\sum_{n=1}^{\infty} \frac{n}{n^2+4}}\)   solution

(d) \(\displaystyle{\sum_{n=2}^{\infty} \frac{\ln n}{n}}\)    solution

(e) \(\displaystyle{\sum_{n=2}^{\infty} \frac{1}{n\ln n}}\)   solution

(f) \(\displaystyle{\sum_{n=1}^{\infty} n^2 e^{-n^3}}\)   solution

(g) \(\displaystyle{\sum_{n=2}^{\infty} \frac{e^{1/n}}{n^2}}\)  solution

(h) \(\displaystyle{\sum_{n=3}^{\infty} \frac{n^2}{e^n}}\)   solution