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limit n tends to infinity.....summation of r/(n^2+n+r)...from r=1 to r=n is???? a)0. b)1/3. c)1/2 d)1 answer is 1/2 ...howww???? plzzz

limit n tends to infinity.....summation of


r/(n^2+n+r)...from r=1 to r=n is????


a)0.  b)1/3.   c)1/2     d)1


answer is 1/2 ...howww???? plzzz

Grade:12th Pass

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
8 years ago
Ans:
L = \lim_{n\rightarrow \infty }\sum_{r = 1}^{n}\frac{r}{n^{2}+r+n}
L = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{r = 1}^{n}\frac{\frac{r}{n}}{1+\frac{r}{n^{2}}+\frac{1}{n}}
\frac{r}{n} = x
\lim_{n\rightarrow \infty }\frac{1}{n} = dx
L = \int_{0}^{1}x.dx = (\frac{x^{2}}{2})_{0}^{1} = \frac{1}{2}
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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