Absolute and conditional convergence practice problems
Absolutely convergent, conditionally convergent, or divergent? The alternating series test and a comparison for |aₙ|. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n + 1} \cdot 2^{- n} 3^{n} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} n}{n^{2} + 1} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n + 1} \cdot 3^{- n} n \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2} + 1} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n + 1} \cdot 2^{- n} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{\sqrt{n}} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{\frac{3}{2}}} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{2 n + 1} \) converge absolutely, converge conditionally, or diverge?
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} \left(n + 1\right)}{n^{3}} \) converge absolutely, converge conditionally, or diverge?