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The Lim 1/(n+r) Problem (TOMATO Subjective 155 with some modifications)

Problem:  Evaluate: \(\lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+…+\frac{1}{n+n})\)

Solution:

As the title suggests the modification of this problem will be, that we will solve a more general series and then use a specific value to arrive at the solution of this problem.

First let us consider the following limit:

\(\lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+…+\frac{1}{n+kn})\)

Observe carefully that using k=1 in this limit, we get the limit that has been asked to evaluate.

 

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