Lim - k -1 n k - n 2- k 2 is equals to -Roorkee 1999-A- 1-2log 2B- log 2C- - - 4D- - - 2

The correct option is A Let I= lim_π→∞∑ _k=1^n k/n^2+k^2=lim_π→∞∑ _k=1^n 1/n ( k/n )/1+ ( k/n )^2 I = ∫_0^1 x/1+x^2dx=1/2 [log (1+x^2)]^1_0 =1/2[log 2] ...

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Byju's Answer

Standard XII

Mathematics

Definite Integral as Limit of Sum

lim π→∞∑ k =1...

Question

$l i m_{π \to \infty} \sum_{k = 1}^{n} \frac{k}{n^{2} + k^{2}}$ is equals to [Roorkee 1999]

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Solution

The correct option is A
$L e t I = l i m_{π \to \infty} \sum_{k = 1}^{n} \frac{k}{n^{2} + k^{2}} = l i m_{π \to \infty} \sum_{k = 1}^{n} \frac{1}{n} \frac{(\frac{k}{n})}{1 + {(\frac{k}{n})}^{2}}$
$I = \int_{0}^{1} \frac{x}{1 + x^{2}} d x = \frac{1}{2} [l o g (1 + x^{2})]_{0}^{1} = \frac{1}{2} [l o g 2]$

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