IGNOU BMTC 133 2026 - Discuss the convergence of the p-series by using the integr

Question

Discuss the convergence of the p-series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}, \ p > 0$ by using the integral test.

20 Jan 2026

Answer :

Word Count : 162

Consider the p-series (\sum_{n=1}^{\infty}\frac{1}{n^{p}}) with (p>0). To apply the integral test, take the function (f(x)=\frac{1}{x^{p}}) for (x\ge 1). This function is positive, continuous, and decreasing on ([1,\infty)) for all (p>0), so the conditions of the integral test are satisfied. We examine the improper integral [ \int_{1}^{\infty}\frac{1}{x^{p}},dx. ] For (p\neq

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