Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

is there a method to evaluate the fourier sums ??

$$ \sum_{n=0}^\infty t^n \sin(nx)= F(x,t) $$

$$ \sum_{n=0}^\infty t^n \cos(nx)= G(x,t) $$

my idea is that i need to use these sums to apply Borel Resummation to the series

$$ \sum_{n=0}^\infty a(n) \cos(nx)= \int_0^\infty dt \, g(t)G(x,t) $$

with $ a(n)= \int_0^\infty dt \, t^{n}g(t) $

share|cite|improve this question
up vote 4 down vote accepted


$$\sum_{n=0}^{\infty} t^n e^{i n x} = \frac{1}{1-t e^{i x}}$$


$$\sum_{n=0}^{\infty} t^n \cos{n x} = \Re{\frac{1}{1-t e^{i x}}}$$

$$\sum_{n=0}^{\infty} t^n \sin{n x} = \Im{\frac{1}{1-t e^{i x}}}$$

When $t \in \mathbb{R}$, then

$$\begin{align}\frac{1}{1-t e^{i x}} &= \frac{1}{1-t \cos{x} - i t \sin{x}} \\ &= \frac{1-t \cos{x} + i t \sin{x}}{(1-t \cos{x})^2 + t^2 \sin^2{x}}\\ &= \frac{1-t \cos{x}}{1-2 t \cos{x} + t^2} + i \frac{t \sin{x}}{1-2 t \cos{x} + t^2} \end{align}$$

share|cite|improve this answer
OK thanks :) .. – Jose Garcia Feb 18 '13 at 11:37

Let $\rm H(x,t) = G(x,t) + i F(x,t)$, then $$ \rm H(x,t) = \sum_{n = 0}^{\infty} \cos(nx) + i\sin(nx) = \sum_{n=0}^{\infty} (te^{ix})^n = \frac{1}{1- e^{ix}}$$

Then $\rm G(x,t) = \mathfrak{Re}(\rm H(x,t)) = \frac{1 - t\cos x}{1 - 2t\cos x + t^2}$ because $\mathfrak{Re}$ commutes with summation.

In the same way, $\rm F(x,t) = \frac{t\sin x}{1 - 2t\cos x + t^2}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.