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Mathematical Analysis Fundamentals by Agamirza Bashirov

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image

and

βn=1π-ππf(x)sinnxdx=-nπ-ππf(x)cosnxdx=-nan.

image

Finally, by Theorem 4.7, Corollary 12.33, and Example 12.15,

n=1man2+bn22=n=1m1nαn2+βn22n=1m1n2n=1m(αn2+βn2)π6-ππf(x)2dx.

image

Therefore, the sequence of partial sums of the series n=1an2+bn2image is bounded and increasing, respectively, and it converges.  

Theorem 12.46

Let fC(-π,π)PS ...

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