7. Which of the following series are convergent? I. \[ 1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}+\cdots \] II. \[ 1-\frac{1}{2}+\frac{1}{3}-\cdots+\frac{(-1)^n}{n}+\cdots \] III. \[ 2+1+\frac{8}{9}+\cdots+\frac{2^n}{n^2}+\cdots \]
Answer is: option3
Solution:
I. This is a \(p\)-series with \(p=2\). Convergent II. This is the alternating harmonic series. Convergent III. \[ \lim_{n\to\infty} a_n= \lim_{n\to\infty}\frac{2^n}{n^2} =\infty \] \[ \lim_{n\to\infty} a_n \ne 0 \] Divergent
