## Example 8.3.2 (a): Function Series Examples

Show that for a fixed

*x (-1, 1)*the series*x*converges pointwise to the function^{n}*f(x) =*.^{1}/_{1-x}Once we *fix* a value of *x* the series (and the proof) is
*exactly* the same as our regular old
Geometric Series. There we used an
*a*, now we use an *x*. But since repetition is helpful
for memorization in long-term "storage", let's repeat the standard proof
for convergence of the geometric series. Let

S_{N}= 1 + x + x^{2}+ ... + x^{N}

be the *N*-th partial sum. Then

x S_{N}= x(1 + x + x^{2}+ ... + x^{N}) = x + x^{2}+ ... + x^{N}+ x^{N+1}

But then

S_{N}- x S_{N}= S_{N}(1 - x) = 1 - x^{N+1}

Thus we have

S_{N}=^{1 - xN+1}/_{1-x}

Now we can take limits as *N* goes to infinity:

- if
|x| < 1the termxgoes to zero^{N+1}- if
|x| > 1the termxdiverges^{N+1}

which finishes the proof.