(a) Find the Taylor series for .
(b) Why does your answer make sense?
(a) Remember the chain rule (link) and be careful with the minus signs.
(b) You've seen the expression before. Where?
a) Let's take some derivatives and evaluate them at 0. We start with
Taking the derivative and remembering to use the chain rule, we get
Each time we take a derivative, we'll get two negative signs, one from the exponent and one from the derivative of the inside function (1 – x). These negative signs will always cancel out.
So we get
Evaluating at 0, we have
and so on. For every n, f(n)(0) = n!.
Now we put this information into the formula and get
A graph confirms that this is reasonable:
b) The expression is the sum of the infinite geometric series with ratio x, where |x| < 1.We just found that the Taylor series for near zero is
1 + x + x2 + x3 + ... + xn + ...
This is the infinite geometric series with ratio x. The condition |x| < 1 sounds like x being "close to zero".