The Ratio and Root Tests    Lecture Video

I. The following three problems are related.  solution

(a) Determine if the series \( \displaystyle{\sum_{n=1}^{\infty} (-1)^{n} \: \frac{1}{\sqrt[3]n}}\) is convergent or divergent.
(b) Determine if the series \( \displaystyle{\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]n}}\) is convergent or divergent.
(c) From parts (a) and (b) conclude if the series \( \displaystyle{\sum_{n=1}^{\infty} (-1)^{n} \: \frac{1}{\sqrt[3]n}}\) is absolutely convergent, conditionally convergent or divergent.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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