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762

Part of 2011 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 762

Source: 1993 Kogakuin Univesity entrance exam/Engineering

11/5/2011
Define a function fn(x) (n=0, 1, 2, ⋯ )f_n(x)\ (n=0,\ 1,\ 2,\ \cdots) by
f0(x)=sin⁡x, fn+1(x)=∫0π2fn′(t)sin⁡(x+t)dt.f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.
(1) Let fn(x)=ansin⁡x+bncos⁡x.f_n(x)=a_n\sin x+b_n\cos x. Express an+1, bn+1a_{n+1},\ b_{n+1} in terms of an, bn.a_n,\ b_n.
(2) Find ∑n=0∞fn(π4).\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).
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