Let f:(0,2)→R be defined as f(x)=log2(1+tan(πx4)). Then limn→∞2n(f(1n)+f(2n)+…………+f(1)) is equal to _ _ _ _ .
Solution
The given limit limn→∞2n(f(1n)+f(2n)+…………+f(1))
=limn→∞(n∑r=1f(rn)).2n
=1∫0f(x).2dx=I
I=21∫0log2(1+tanπx4)dx
⇒I=21∫0log2(1+tanπ(1–x)4)dx=21∫0log2(1+tan(π4–πx4))dx
⇒I=21∫0log2(1+1–tanπx41+tanπx4)dx=21∫0log2(21+tanπx4)dx
⇒I=21∫0[log22–log2(1+tanπx4)]dx=21∫0[1–log2(1+tanπx4)]dx
⇒I=2–I
⇒I=1