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Example 8.3.2 (a): Function Series Examples

Show that for a fixed x (-1, 1) the series xn converges pointwise to the function 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

SN = 1 + x + x2 + ... + xN

be the N-th partial sum. Then

x SN = x(1 + x + x2 + ... + xN) = x + x2 + ... + xN + xN+1

But then

SN - x SN = SN(1 - x) = 1 - xN+1

Thus we have

SN = 1 - xN+1/1-x

Now we can take limits as N goes to infinity:

  • if |x| < 1 the term xN+1 goes to zero
  • if |x| > 1 the term xN+1 diverges

which finishes the proof.

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