Polynomial approximation

[Jimmy84]

Homework Statement

Obtain the Taylor polynomials Tnf(x) as indicated. In each case, it
is understood that f(x) is defined for a11 x for which f(x) is meaningful.


Problem one
Tn = (a^x) = sigma from k = 0 to n of ((log a)^k)/k! x^k

Problem two
Tn = (1/(1+x)) = sigma from k = o to n of (-1)^k x^k

Homework Equations

The Attempt at a Solution

Im totally lost and I don't know where to start can anyone help me please?

 

[HallsofIvy]

Did you really have to use "a11" when you have a perfectly good "l" key?

They Taylor series for function f(x) is $$\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$$ where $f^{(n)}(0)$ indicates the nth derivative of f evaluated at 0.

(Actually, that is the "Taylor series at 0" or "MacLaurin" series.)

So for $a^x$ you only need to find the derivatives of $a^x$ and evaluate at x= 0. That can be done most efficiently by writing $a^x= e^{ln a^x}= e^{x ln a}$.

It is also true that any power series equal to a given function is a Taylor series. If you remember that the sum of the geometric series $\sum_{n=0}^\infty r^n$ is equal to 1/(1- r), then the Taylor series for 1/(1+x) should be obvious.