# Very very basic taylor series problem

1. Sep 22, 2007

### PTTB

1. The problem statement, all variables and given/known data

Consider f(x) = 1 + x + 2x^2+3x^3.
Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function

2. Relevant equations

3. The attempt at a solution

This is the first time that I have seen Taylor series and I am totally lost on how to do it, I have looked all around the internet for some help and I just don't have a clue on where to start. Any Help would be appreciated
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 22, 2007

### Gib Z

Well start from the definition of a taylor series:
The taylor series of a function centered around a is given by
$$\sum_{n=0}^{\infty} f^n (0) \frac{(x-a)^n}{n!}$$ where f^n is the nth derivative of f, not the nth power. The 0th derivative is just the original function.

So it wants you to find 2 series, one with center 0, or a=0, and the other with centre a=x_0. Use the formula to get the taylor series centered around those points, and then since it only wants a linear function, only use the first 2 terms, ie the constant and the x terms.

3. Sep 22, 2007

### PTTB

thanks a ton, i think im starting to get this stuff, you were a huge help

i got 1+x and 1+x-2x_o^2-6x_0^3+(4x_0)x+(9x^2_0)x im pretty sure thats rite, thanks a ton

4. Sep 23, 2007

### HallsofIvy

Staff Emeritus
Actually, it makes little sense to say "Taylor series approximation". The Taylor series is exact. It is the "Taylor polynomial" that is approximate and it is my guess that this is what is intended. Of course, the Taylor Polynomial that gives a linear approximation is just the 1st order approximation: f(x0)+ f '(x0)(x- x0)- and that's just the tangent line approximation.
Since $f(x_0)= 1+ x_0+ 2x_0^2+ 3x_0^3$ and $f '(x_0)= 1+ 4x_0+ 9x_0^2$, the tangent line approximation at $x_0$ is $1+ x_0+ 2x_0^2+ 3x_0^3+ (1+ 4x_0+ 9x_0^3)(x- x_0)$. After multiplying out the last term, that gives exactly what you have.

5. Sep 23, 2007

### genneth

*Grumble*
*Mutterings about analytic functions and radius of convergence*
*Grumble*