Let f : (0 ,2 ) R be defined as f (x )= log_2(1+ tan ( πx4 )) Then, limn 2n(f ( 1n )+f ( 2n )+ … .+f (1 )) is equal to ______

Question

Accepted Answer

Correct answer: 1. (1) E=2 limn 1nf ( rn ) E= 2 ln2 ln(1+ tan πx4) dx Replacing x→1 –x E= 2 ln2 ln(1+ tan 4(1-x)) dx E= 2 ln2 ln(1+ tan( 4- 4x)) dx E= 2 ln2 ln(1+ 1+ tan 4x1+ tan 4x) dx E= 2 ln2 ln ( 21+ tan πx4 ) dx E= 2 ln2 ( ln2- ln(1+ tan πx4)) dx Equation (i) + (ii) E=1

$Let f : (0, 2) \to R be defined as f (x) = log_{2} (1 + tan (\frac{πx}{4})) Then, lim n \to \infty \frac{2}{n} (f (\frac{1}{n}) + f (\frac{2}{n}) + \dots . + f (1)) is equal to ______$

Solution

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