The Ratio and Root Tests Worksheet with Answer Key

What's on the worksheet

1. Use the ratio test to decide whether each series converges absolutely or diverges.
   \sum_{n=1}^{\infty} \dfrac{n^2\, 3^n}{n!}\sum_{n=1}^{\infty} \dfrac{n!}{1 \cdot 4 \cdot 7 \cdots (3n - 2)}\sum_{n=1}^{\infty} \dfrac{n!}{100^n}\sum_{n=1}^{\infty} \dfrac{(-2)^n\, n!}{n^n}\sum_{n=1}^{\infty} \dfrac{e^{2n}}{n^5\, 7^n}
2. Use the root test to decide whether each series converges absolutely or diverges.
   \sum_{n=1}^{\infty} \left(\dfrac{4n - 1}{3n + 5}\right)^n\sum_{n=1}^{\infty} \dfrac{3^n\, n^4}{n^n}\sum_{n=1}^{\infty} \left(\sqrt[n]{n} - \dfrac{1}{2}\right)^n\sum_{n=1}^{\infty} \dfrac{(-5)^n}{n^2\, 2^{n+3}}
3. For each series, show that the ratio test gives L = 1.
4. Let S be the sum of the series below, and S_N its Nth partial sum.
5. Suppose a_n > 0 for every n and [n]a_n 13.
6. Consider the series below, with sum S and Nth partial sum S_N.