7. find the values of p for which the series ∑(n = 1 to ∞) 1 / n²ᵖ converges. a. p<0 b. p≥1 c. p<2 d. p>1/2…

Explanation:

Step1: Recall p - series test

The p - series is given by $\sum_{n = 1}^{\infty}\frac{1}{n^k}$, and it converges if $k>1$ and diverges if $k\leq1$. In our series $\sum_{n = 1}^{\infty}\frac{1}{n^{2p}}$, we let $k = 2p$.

Step2: Apply convergence condition

For the series $\sum_{n = 1}^{\infty}\frac{1}{n^{2p}}$ to converge, we need $2p>1$ according to the p - series test. Solving the inequality $2p>1$ for $p$, we divide both sides by 2: $p>\frac{1}{2}$.

Answer:

D. $p>\frac{1}{2}$