Evaluate the series 1+1/9+1/(25)+1/(49)+cdots

miniliv4 2022-10-02 Answered
Evaluate the series
1 + 1 9 + 1 25 + 1 49 +
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Answers (2)

beshrewd6g
Answered 2022-10-03 Author has 12 answers
One approach is as follows:
Let f ( t ) denote a triangle wave with the precise form described here (we'll take L=1). As you can see in the link, f can be expanded into its Fourier series as
f ( t ) = 8 π 2 n = 1 ( 1 ) n 1 ( 2 n 1 ) 2 sin ( ( 2 n 1 ) π t )
Now, just plug t = 1 2 into both sides of the equation above.
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Denisse Fitzpatrick
Answered 2022-10-04 Author has 3 answers
Hint: On way to evaluate your series is using Parseval theorem. At first you need to find the fourier series of a function like f ( x ) = x in [ π , π ] and then applying Parseval theorem.
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