Investigate the alternating series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$.
A) The p- series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ converges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges absolutely.
B) The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ converges conditionally.
C) The ratio test gives $\mathrm{r}=1$, so the series $\sum_{\mathrm{n}=1}^{\infty} \frac{(-1)^{\mathrm{n}+1}}{\mathrm{n}^{2}}$ diverges.
D) The p-series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ diverges, so the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$ diverges.

Question

Quizplus · Accepted Answer

The series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ is a convergent p-series with $p=2>1$, so the alternating series converges absolutely.

Investigate the Alternating Series ∑n=1∞(−1)n+1n2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}∑n=1∞​n2(−1)n+1​ A) the P- Series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^{2}}∑n=1∞​n21​

Investigate the Alternating Series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$
A) the P- Series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$