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Series convergence tests practice problems

Divergence, p-series, integral, comparison and ratio tests: which one, and why. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{2}}} \) converge or diverge?

Problem 7.16easyNot checked

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{5 n + 3} \) converge or diverge?

Problem 7.17easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{3^{n}}{n!} \) converge or diverge?

Problem 7.18easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n^{2} \) converge or diverge?

Problem 7.19easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{n + 1}{n^{3} + n + 1} \) converge or diverge?

Problem 7.20easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{5^{n}}{n!} \) converge or diverge?

Problem 7.21easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{n + 1}{5 n + 3} \) converge or diverge?

Problem 7.22easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{3}} \) converge or diverge?

Problem 7.23easyNot checked

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) converge or diverge?

Problem 7.24easyNot checked

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{2^{n}}{n!} \) converge or diverge?

Problem 7.25easy✓ Every equation proved

Does \( \displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln{\left(n \right)}} \) converge or diverge?

Problem 7.26easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2}} \) converge or diverge?

Problem 7.27easyNot checked

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{n + 1}{n^{2} + n + 1} \) converge or diverge?

Problem 7.28easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{6^{n}}{n!} \) converge or diverge?

Problem 7.29easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{n^{3} + n + 1} \) converge or diverge?

Problem 7.30easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n \) converge or diverge?

Problem 7.76easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4^{n}}{n!} \) converge or diverge?

Problem 7.77easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{3 n + 1}{n^{3} + n + 1} \) converge or diverge?

Problem 7.78easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{2 n + 3} \) converge or diverge?

Problem 7.79easy✓ Every equation proved

Does \( \displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln{\left(n \right)}^{2}} \) converge or diverge?

Problem 7.80easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{3 n + 1}{n + 3} \) converge or diverge?

Problem 7.81easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{5^{n}}{n^{3}} \) converge or diverge?

Problem 7.82easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n^{3} \) converge or diverge?

Problem 7.83easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} 4^{- n} n^{2} \) converge or diverge?

Problem 7.84easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{2^{n}}{n} \) converge or diverge?

Problem 7.85easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \) converge or diverge?

Problem 7.86easyNot checked

Does \( \displaystyle \sum_{n=1}^{\infty} 2^{- n} n^{3} \) converge or diverge?

Problem 7.87easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{5^{n}}{n} \) converge or diverge?

Problem 7.88easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} 4^{- n} n^{3} \) converge or diverge?

Problem 7.89easy✓ Every equation proved

Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{n^{2} + n + 1} \) converge or diverge?

Problem 7.90easy✓ Every equation proved