With S_N(x) = 1 + x + x² + ... + x^N we computed and simplified S_N(x) - x S_N to show that the geometric series converges to ¹/_1-x, so technically we're done. However, that proof only works for this particular series and is not useful as an example of general Taylor series theory. Therefore, we will come up with an alternate proof, more in line with our current topic.

We know - assuming the assumptions of Taylor's theorem are satisfied (which they are) - that:

|f(x) - | = |R_n+1(x)|

where I can pick my preferred form of the remainder. Let's pick the Lagrange form, i.e.:

R_(n+1)(x) =

for some t between x and c. But for our given function we have

f ⁽ⁿ⁺¹⁾(t) = ^(n+1)!/_{(1-t)ⁿ⁺²}

so that

|R_(n+1)(x)| =