# Infinite Series - Finding the 10th derivative

• carlodelmundo
In summary, the conversation is discussing the use of Taylor series to find derivatives of functions. The problem at hand is finding the value of f10(3), the tenth derivative of f at x = 3. The first step is to take the derivative of the given Taylor series ten times and then substitute x=3. This results in the term (1/2)^10 or 9.765 * 10^-4. The significance of this problem is that it highlights the importance of considering both directions when using Taylor series - finding derivatives from a Taylor series and finding a Taylor series from a function to find derivatives. An example of this is given with the function f(x) = 1/(1-x) and finding the eighteenth

## Homework Statement

Let T(x) = $$\sum^{\infty}_{k=0}$$ $$\frac{1}{2^k}$$ $$\frac{(x-3)}{k!}$$k be the Taylor series for a function f. What is the value of f10(3), the tenth derivative of f at x = 3?

## The Attempt at a Solution

I have a very small idea of actually starting this problem. Can I just derive the infinite series by taking derivatives of $$\frac{1}{2^k}$$ $$\frac{(x-3)}{k!}$$k 10 times?

Basically, what is the first step in this problem?

I'm sure you mean (1/2^k)*(x-3)^k/k!. Yes, think about taking the derivative 10 times and then putting x=3. Only one term will survive. What is it? Practice by thinking about x^k. If you take 10 derivatives of that and then put x=0 you will almost always get 0 unless k has a particular value. What is it?

Thanks I got (1/2)^10 or 9.765 * 10^-4.

You're right about that one term. When k = 10, then its 0^0, or 1. Thanks for the tips! Much appreciated!

I just want to emphasize the significance of the problem.

It's too easy to let your thinking run in only one direction -- you compute derivatives so that you may find the Taylor series to your function.

But the other direction is also important -- sometimes we want to find the derivative, and it turns out that we can compute the Taylor series easily. So, we compute the Taylor series so that we may find a derivative!

For example, suppose that we have

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

and we want to know the eighteenth derivative:

$$f^{(18)}(0) = \, ?$$