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Question

Real Analysis Question on Sequences and Series

Consider the series n=1nm(11+1np)\sum_{n=1}^{\infty} n^m \left(\frac{1}{1 + \frac{1}{n^p}}\right) where mm and pp are real numbers. Under which of the following conditions does the above series converge?

A

m>1m > 1.

B

0<m<10 < m < 1 and p>1p > 1.

C

0<m10 < m \leq 1 and 0p10 \leq p \leq 1.

D

m=1m = 1 and p>1p > 1.

Answer

m>1m > 1.

Explanation

Solution

The correct option is (A): m>1m > 1.