limitssequences-and-series
Say we have a series $a_0+a_1+a_2+…+a_n+…$ that has the partial sums
$$S_0 = a_0$$
$$S_1 = a_0+a_1$$
$$…$$
$$S_n = a_0+a_1+…+a_n$$
The series converges if $\lim_{n\to\infty} S_n = S$, where S is the sum of the series, i.e. $S = \sum_{j=0}^\infty a_j$.
Why is the limit the sum of the series?
This community wiki solution is intended to clear the question from the unanswered queue.
We define convergence of a series as follows:
The series $\displaystyle \sum_{k = 1}^\infty a_k$ converges if and only if its sequence of partial sums $\displaystyle S_n = \sum_{k = 1}^n a_k$ converges.
When stating definitions, authors write "if" instead of "if and only if" as mentioned in the comments.
I can see two problems.
First off, your Fourier coefficients as given are only half as big as they should be to match your Series1 data. Perhaps you used the even nature of the target function so you only integrated over half the interval but then forgot to multiply by $2$ to take into account the half of the integral you didn't do.
Second, your value for $a_3$ is very high. I didn't actually look at your spreadsheet; here's my Matlab program:
% fouriertest.m
npts = 300;
x = linspace(-2,2,npts);
xgap = [-2 -1 -1 -1 0 0 0 1 1 1 2];
ygap = [20 20 NaN 10 0 NaN 0 10 NaN 20 20];
y0 = 25/2*ones(size(x));
y1 = (-20/pi-40/pi^2)*cos(pi/2*x);
y2 = -20/pi^2*cos(pi*x);
y3 = (20/3/pi-40/pi^2)*cos(3*pi/2*x);
plot(xgap,ygap,'b-',x,y0,'r-',x,y0+y1,'g-',x,y0+y1+y2,'m-', ...
x,y0+y1+y2+y3,'c-');
title('Partial Sums of Fourier Series');
xlabel('x');
ylabel('y');
legend('Series1','S_0','S_1','S_2','S_3','Location','southeast');
And its output:
BTW, your expression for the Fourier series has a typo which I an gonna fix.
Best Answer
By definition, indeed.
There is no ground to speak of the "sum" of a series, say $1 - 1 + 1 - 1 + \cdots$, for one may argue that $(1-1) + (1-1) + \cdots = 0$ and another may argue that $1 - (1-1) - (1-1) - \cdots = 1$, a blatant contradiction.
To talk of the sum of an infinite series in a meaningful way, a covenant has been made, which is precisely the definition interesting you.