Simple mathematics (Fourier series)

[johnchau123]

As I have proved the two Fourier series above, what value should I substitute for x in order to find S1?

Thanks!

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[Kral]

Johnchau123,
If I understand your question correctly, S1 is not a function of x. Its summation is 1.64493406684822643637 according to the CRC Standard Mathematical Tables.

 

[johnchau123]

Hi, Kral, I can get your point.

Actually what I mean is as there are sine and cosine function of x in the series.

So, I would like to ask what value of x should be used so that some terms eliminate, say, if i would like to evaluate Σ(-1)^n/(n^2), what I should do is to substitute 2pi into the first series with range pi < x < 3pi.

 

[picagor]

this is very good site for mathematics

 

[johnchau123]

Hi, everyone, I get the solution of this question.

I put x = 1 into the Fourier series which is available for 1 < x < 2. As the Fourier series is continuous on (1,2) and its derivative is also continuous on (1,2), so, we apply the mean value theorem and can find at x = 1, x² = 5/2 and the sum of this series is pi²/6.

 

[dedede]

in the first equation, let x=2pi, and for the second, let x=1.5