Consider the following function described as a Fourier Sine Series:
$$ F(t, x) = \sum_{n = 1}^{\infty} C_{n}e^{- \alpha n^{2} \pi^{2} t}sin(n \pi x)$$
I want to know whether using all the terms of this sum, going to infinity, it's possible to that sum of sines describe a straight line (constant, crescent or decrescent).
If so, how I could give proof that it can actually happen?
Yes for example: The Fourier series $$f(x)=\frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-1)^n)}{n} \sin nx,$$ represents $$f(x)=\frac{x}{\pi}, -\pi<x<\pi$$ See the figure for the sum uptp $n=1000$. See the figure below: