The Direct and Limit Comparison Tests Worksheet with Answer Key

What's on the worksheet

1. Decide whether each series converges or diverges by the direct comparison test.
   \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 7n}\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n\,3^n}\displaystyle\sum_{n=2}^{\infty} \dfrac{\ln n}{n}\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n}
2. Decide whether each series converges or diverges by the limit comparison test.
   \displaystyle\sum_{n=1}^{\infty} \dfrac{3n^2 - 1}{n^4 + 5n}\displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n} + 1}{2n - 1}\displaystyle\sum_{n=1}^{\infty} \left(e^{1/n} - 1\right)\displaystyle\sum_{n=1}^{\infty} \dfrac{2n + \ln n}{n^3 + 1}
3. Decide whether each series converges or diverges.
   \displaystyle\sum_{n=2}^{\infty} \dfrac{1}{(\ln n)^2}\displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n}}{e^{\sqrt{n}}}\displaystyle\sum_{n=1}^{\infty} \left(\sqrt{n^2 + 1} - n\right)
4. Each argument below has a flaw.
5. Let S = _n=1^∞ 1n³ + 1, and let S_N be its Nth partial sum.
6. Suppose a_n > 0 for every n and _n=1^∞ a_n converges.