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  • Evaluating      <img alt="\mathrm{\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}}\right)}" src="https://entra

Evaluating      \mathrm{\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}}\right)}

Option: 1

\sinh ^{-1} 1


Option: 2

\sinh ^{-1} 2


Option: 3

0


Option: 4

None


Answers (1)

best_answer

The sum can be manipulated into a Riemann sum
\mathrm{\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{1}{\sqrt{n^2+k^2}}=\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{1}{\sqrt{1+\frac{k^2}{n^2}}} \cdot \frac{1}{n} \equiv \lim _{n \rightarrow \infty} \sum_{k=1}^n f\left(x_k\right) \Delta x }
which converges to the integral
\mathrm{\longrightarrow \int_0^1 \frac{d x}{\sqrt{1+x^2}}=\sinh ^{-1}(1) }

Posted by

Kuldeep Maurya

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