6. find the values of p for which the series ∑(n = 1 to ∞) 1 / 5n^p diverges. a. p < 7 / 5 b. p ≥ 1 c. p ≤ 1…

Explanation:

Step1: Recall p - series test

A p - series is of the form $\sum_{n = 1}^{\infty}\frac{1}{n^{p}}$, and it converges if $p>1$ and diverges if $p\leq1$. The given series is $\sum_{n = 1}^{\infty}\frac{1}{5n^{p}}=\frac{1}{5}\sum_{n = 1}^{\infty}\frac{1}{n^{p}}$. The constant factor $\frac{1}{5}$ does not affect the convergence or divergence of the series.

Step2: Determine divergence condition

According to the p - series test, the series $\sum_{n = 1}^{\infty}\frac{1}{n^{p}}$ diverges when $p\leq1$.

Answer:

C. $p\leq1$