By the definition of definite integral, the value of limn→∞(1415+n5+2425+n5+3435+n5+−−−+n4n5+n5) is

limn→∞1415+n5+2425+n5+3435+n5+.......+n4n5+n5 limn→∞∑r=1nr4r5+n5=limn→∞1n∑r=1n(r/n)4(rn)5+1=∫01x2x5+1dx=15[log(x5+1)]01=15log2

By the definition of definite integral, the value of limn→∞(1415+n5+2425+n5+3435+n5+−−−+n4n5+n5) is

limn→∞1415+n5+2425+n5+3435+n5+.......+n4n5+n5 limn→∞∑r=1nr4r5+n5=limn→∞1n∑r=1n(r/n)4(rn)5+1=∫01x2x5+1dx=15[log(x5+1)]01=15log2

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By the definition of definite integral, the value of $\lim_{n \to \infty} (\frac{1^{4}}{1^{5} + n^{5}} + \frac{2^{4}}{2^{5} + n^{5}} + \frac{3^{4}}{3^{5} + n^{5}} + - - - + \frac{n^{4}}{n^{5} + n^{5}})$ is

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$\frac{1}{5} \log 2$

$\frac{1}{4} \log 3$

$\frac{1}{3} \log 2$

log 1

answer is A.

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Detailed Solution

$\lim_{n \to \infty} \{\frac{1^{4}}{1^{5} + n^{5}} + \frac{2^{4}}{2^{5} + n^{5}} + \frac{3^{4}}{3^{5} + n^{5}} + ....... + \frac{n^{4}}{n^{5} + n^{5}}\} \lim_{n \to \infty} \sum_{r = 1}^{n} \frac{r^{4}}{r^{5} + n^{5}} = \lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} \frac{{(r / n)}^{4}}{{(\frac{r}{n})}^{5} + 1} = \int_{0}^{1} \frac{x^{2}}{x^{5} + 1} d x$
$= \frac{1}{5} {[\log (x^{5} + 1)]}_{0}^{1} = \frac{1}{5} \log 2$