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

miniliv4

miniliv4

Answered question

2022-10-02

Evaluate the series
1 + 1 9 + 1 25 + 1 49 +

Answer & Explanation

beshrewd6g

beshrewd6g

Beginner2022-10-03Added 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.
Denisse Fitzpatrick

Denisse Fitzpatrick

Beginner2022-10-04Added 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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