Absolute Convergence: Ratio and Root Tests    Lecture Video

1. Determine whether each of the following series is absolutely convergent, conditionally convergent or divergent.

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

(b) \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n-1} \: \frac{1}{2n+3}}\)   solution

(c)  \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n-1} \: \frac{1}{n}}\)   solution

(d) \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n+1} \: \frac{2^n-1}{3^n-1}}\)   solution

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

(f) \(\displaystyle{\sum_{n=1}^{\infty} \frac{n!}{{100}^n}}\)  solution

(g) \( \displaystyle{\sum_{n=1}^{\infty} (-1)^n\frac{1}{\sqrt[3]{n}}}\)  solution

(h) \(\displaystyle{\sum_{n=1}^{\infty} (-1)^{n-1}\,\frac{2n+5}{3n+1}}\)  solution

2. Determine whether each of the following series is convergent or divergent.

(a) \(\displaystyle{\sum_{n=1}^{\infty} \frac{(-2)^n}{n}}\)  sol I   sol II

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

(c) \(\displaystyle{\sum_{k=1}^{\infty} \frac{3}{k!}}\)    solution

(d) \(\displaystyle{\sum_{n=1}^{\infty} \left( \frac{n^2+1}{3n^2+2}\right)^n}\)   solution

(e) \(\displaystyle{\sum_{m=1}^{\infty} \frac{m}{e^m}}\)    solution

(f) \(\displaystyle{\sum_{n=1}^{\infty} \left( \frac{5n^2-2}{4n^2+3}\right)^n}\)    solution

(g) \(\displaystyle{\sum_{n=1}^{\infty} \frac{n!}{n^n}}\)  solution

(h) \(\displaystyle{\sum_{n=1}^{\infty}(-1)^{n-1} \: \frac{7^{n}}{5^n\, n^3}}\)  solution

(i) \(\displaystyle{\sum_{n=3}^{\infty} 4 \left(1+ \frac{1}{n}\right)^{n^2}}\)    solution