IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

amulye

$\mathop{\lim}\limits_{{n} \to {\infty}}$ { 1^5/2 +2^5/2+3^5/2+........... +n^5/2 } / n^7/2 is =?

akku

$\mathop{\lim}\limits_{{n} \to {\infty}}$ $\sum_{r=1}^{n}$ r=1 r^5/2/n^7/2= $\mathop{\lim}\limits_{{n} \to {\infty}}$ $\sum_{r=1}^{n}$ (r/n)^7/2.(1/r)

= $\int_{0}^{1}$ x^7/2 dx =2/9x^9/2+c=2/9

allamraju

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Ans is 2/7.It is of the form lim(1/n) $\sum_{r=1}^{n}$ f(r/n) as n tends to infinity where f(x)=x^5/2.Thus,the limit is $\int_{0}^{1}$ x^5/2dx=2/7

@akku,U made a mistake.

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Ans is 2/7.It is of the form lim(1/n)f(r/n) as n tends to infinity where f(x)=x5/2.Thus,the limit is x5/2dx=2/7

@akku,U made a mistake.

Ans is 2/7.It is of the form lim(1/n) $\sum_{r=1}^{n}$ f(r/n) as n tends to infinity where f(x)=x^5/2.Thus,the limit is $\int_{0}^{1}$ x^5/2dx=2/7