Expand in Fourier series of $f(x) = \sin x$ for $0
. Deduce the result \[ \frac1{1 \cdot 3} - \frac{1}{3\cdot5} +\frac{1}{5\cdot 7} - \cdots = \pi-\frac{2}{4}. \] Obtain half range sine series for $f(x)$ in $0
and deduce the series \[\sum\frac{1}{n^2} = \frac{{\pi^2}}6.\]
Fourier and half range series for $\sin x$
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fourier-analysis
fourier-series
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2Perhaps you should explain what you have tried already? – 2012-07-20
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0I tried to fix the source and tags — whatever you typed wasn't displaying at all on my end, so I tried to put everything in LaTeX. Please check for errors. [It's a good idea to try to learn the rudiments of LaTeX for the purpose of asking questions here, by the way.] In particular, I couldn't tell what you were trying to write in that first series. It doesn't look right to me as it is. – 2012-07-20
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0Presumably, the term $\frac 1 {3\cdot 7}$ is meant to be $\frac 1 {5\cdot 7}$, or else I don't see the pattern to the series. – 2012-07-20
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0@ThomasAndrews Change made. – 2012-07-20
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0@ThomasAndrews I hv the same question as it is written....May be there is a printing mistake...Please can you provide me the steps of the solution taking 1/5.7 if you think its correct....and the steps for the second part too....please.... – 2012-07-20
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0Your LHS is 0.783731515 (considering first 150 terms) while your RHS is 2.641592654. Please check . Your RHS should be $\frac{\pi}{4}- \frac{1}{2}$ - for the first series – 2012-07-21