# Exploring the Upper Bound of f^{n+1}(x) for x in [-1/2, 1/2]

• Punkyc7

#### Punkyc7

Let f = ln($\frac{1}{1-x}$)

show that if x $\in$ [-1/2 , 1/2] then

|f$^{n+1}$(x)| <= 2$^{n + 1}$ * n!

I am having a hard time seeing how 2$^{n + 1}$ * n! comes into play.

I have that the taylor series for f is $\Sigma$ $\frac{x^n}{n}$

If a take a derivative it becomes x^(n-1) and if I plug anything on the interval it is less than one. I am thinking that I did this wrong because of how big that upper bound is/.

You're currently using ##n## for two different purposes: the order of the derivative and the index of the sum. I strongly suggest using a different letter for one of these. For example, ##f(x) = \sum_{k=1}^{\infty} x^k/k##. And you want to show that ##|f^{(n+1)}(x)| \leq 2^{n+1}n!##. (I assume your exponent ##n+1## means the ##n+1##'st derivative.)

Try starting with ##n=0##. Can you show that the inequality is true in that case?

No one said you have to use the Taylor's series. Why don't you take 2 or 3 of derivatives directly and see if a pattern emerges. That could lead you to a proof by induction.