f(x)=log2(1+tan(πx4))

limn2n(f(1n)+f(2n)++f(1))=?

Let f:(0,2)R be defined as f(x)=log2(1+tan(πx4)). Then limn2n(f(1n)+f(2n)++f(1)) is equal to _ _ _ _ .

Solution

The given limit limn2n(f(1n)+f(2n)++f(1))

=limn(nr=1f(rn)).2n

=10f(x).2dx=I

I=210log2(1+tanπx4)dx

I=210log2(1+tanπ(1x)4)dx=210log2(1+tan(π4πx4))dx

I=210log2(1+1tanπx41+tanπx4)dx=210log2(21+tanπx4)dx

I=210[log22log2(1+tanπx4)]dx=210[1log2(1+tanπx4)]dx

I=2I

I=1

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