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x^2+x is expressed as Fourier series in (-3,3), then the Fourier series at I = -3
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Step-by-step explanation:
Arithmetico-Geometric Series and Polylogarithms
Date: 07/06/2006 at 16:27:43
From: Karthik
Subject: What is the sum of the following infinite series
In my research about microprocessor temperature behaviour, I
encountered a series similar to the one below:
e^(-x) + 1/9 * e^(-3x) + 1/25 * e^(-5x) + 1/49 * e^(-7x) + ...
Is there a closed form expression for the sum of this series? I find
the combination of the geometric series e^-(2n+1)x and the series
1/(2n+1)^2 to be the most confusing part.
1) Using Fourier series expansion of a saw-tooth waveform, I could
derive the following relation:
1 + 1/9 + 1/25 + 1/49 + ... = pi^2 / 8
2) The given series can be bounded from above by the geometric series
(as every individual term is larger than that of the given series):
e^(-x) + e^(-3x) + e^(-5x) + ... . Using the formula for the infinite
sum of a geometric series, this implies that the series sum is less
than or equal to e^(-x) / (1 - e^(-2x)).
3) Similarly, the series can be bounded from below by the geometric
series e^(-x)/e + e^(-3x)/e^3 + e^(-5x)/e^5 + ... as every term in
it is smaller than that of the given series. Hence, the series sum
is greater than or equal to e^-(x+1) / (1 - e^-2(x+1)).
Thus, I am able to bind the series sum between f(x) and f(x+1) where
f(x) = e^(-x) / (1-e^(-2x)). However, I am not able to close the gap
further and find an exact expression for the sum. Could you please
help me with it? Thank you.