I am having trouble with the following question:
Integrate the Taylor series:
$\sum_{i=1}^\infty \frac{d^if(x)|_{x_0}}{dx}(x-x_0)^i$
I know that we can represent this function as the sum of integrals under some condition?
$f(x)=\int \sum_{i=1}^\infty \frac{d^if(x)|_{x_0}}{dx}(x-x_0)^i \underset{?}{=} \sum \int d^i…$
But I don't understand how to finish this calculation.
Can any one give me a hand with this?
Best Answer
Note you omitted $i!$ from the denominator of the generic term.
You've also omitted the constant term (the term where $i=0$).
In the generic term $\frac{f^{(i)}(x_0)(x-x_0)^i}{i!}$, the factor $\frac{f^{(i)}(x_0)}{i!}$ is a constant, so it comes out of the integral as a multiplier; the only part you need to worry about integrating is $(x-x_0)^i$.
And you will have an arbitrary additive constant as well, unless this is a definite integral.