## Example 8.3.2 (b): Function Series Examples

This is very interesting: adding up infinitely many "wavy" *sine*
functions is - supposedly - going to be equal to a linear function with
slope *-1*. Indeed, that seems to be correct, as the graphs
of two partial sums below show:

The 10-th partial sum

The 100-th partial sum

The proof will be an application of trig identities and a previous integration-by-parts result, but it should be clear how we need to proceed (just as in the geometric series example):

- we need to consider the N-th partial sum
*S*_{N}(x) = sin(x) + 1/2 sin(2x) + ... + 1/N sin(Nx) - we need to find a "closed form" for
*S*that does not involve a summation (or ... dots)_{N}(x) - we need to take the limit as N goes to infinity of
*S*, using the closed form to evaluate the limit_{N}(x)

The difficult step is the second one, finding a closed form for the partial sums. We will need a technical lemma for that:

## Sum of

ForcosLemmax # 2kwe have

Assuming this lemma to be true, we also need to get *1/n sin(nx)*
involved. But that is easy:

But then, putting things together:

We can rewrite the integral on the right (and evaluate the last part) to get:

where is a continously differentiable function. But we have shown previously that

f(x) sin(nx) dx = 0

Therefore:

which is what we wanted to show.

By the way, our statement only applies for
*0 < x < *. Where in our
proof did we use this restriction?

So it remains to prove our technical lemma ...