If f is a periodic function with period T & [.] denotes GIF, then, Lt=[f(x+T)f(x+T)]+[22f(x+2T)f(x+2T)]+[32f(x+3T)f(x+3T)]+……+[n2f(x+nT)f(x+nT)]n3

since T is a periodic function with f=(f(x))f(x)limn→∞∑n2n3=(f(x))f(x)limn→∞n(n+1)(2n+1)6n3=(f(x))f(x)3

If f is a periodic function with period T & [.] denotes GIF, then, Lt=[f(x+T)f(x+T)]+[22f(x+2T)f(x+2T)]+[32f(x+3T)f(x+3T)]+……+[n2f(x+nT)f(x+nT)]n3

since T is a periodic function with f=(f(x))f(x)limn→∞∑n2n3=(f(x))f(x)limn→∞n(n+1)(2n+1)6n3=(f(x))f(x)3

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If f is a periodic function with period T & [.] denotes GIF, then, $L t = \frac{[f (x + T)^{f (x + T)}] + [2^{2} f (x + 2 T)^{f (x + 2 T)}] + [3^{2} f (x + 3 T)^{f (x + 3 T)}] + \dots \dots + [n^{2} f (x + n T)^{f (x + n T)}]}{n^{3}}$

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$\frac{(f (x))^{f (x)}}{2}$

$\frac{(f (x))^{f (x)}}{4}$

$\frac{2 (f (x))^{f (x)}}{3}$

$\frac{(f (x))^{f (x)}}{3}$

answer is D.

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Detailed Solution

since T is a periodic function with f

$= (f (x))^{f (x)} [lim_{n \to \infty} \frac{\sum n^{2}}{n^{3}}]$

$= (f (x))^{f (x)} [lim_{n \to \infty} \frac{n (n + 1) (2 n + 1)}{6 n^{3}}]$

$= \frac{(f (x))^{f (x)}}{3}$

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If f is a periodic function with period T & [.] denotes GIF, then, Lt=[f(x+T)f(x+T)]+[22f(x+2T)f(x+2T)]+[32f(x+3T)f(x+3T)]+……+[n2f(x+nT)f(x+nT)]n3

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If f is a periodic function with period T & [.] denotes GIF, then, $L t = \frac{[f (x + T)^{f (x + T)}] + [2^{2} f (x + 2 T)^{f (x + 2 T)}] + [3^{2} f (x + 3 T)^{f (x + 3 T)}] + \dots \dots + [n^{2} f (x + n T)^{f (x + n T)}]}{n^{3}}$