1. ## Power series representation

Here's the question:

Find a power series representation for the function and determine the
interval of convergence.

f(x) = 1/(x + 10)

2. Originally Posted by Undefdisfigure
Here's the question:

Find a power series representation for the function and determine the
interval of convergence.

f(x) = 1/(x + 10)
we know that $\frac 1{1 - x} = \sum_{n = 0}^{\infty} x^n$ for $|x| < 1$

now note: $\frac 1{x + 10} = \frac 1{10} \cdot \frac 1{(x/10) + 1} = \frac 1{10} \cdot \frac 1{1 - (-x/10)}$

now continue

3. Thanks Jhevon but before I even saw your answer I found the right method in the book and solved the power series representation and found that the interval of convergence is (-10, 10).

4. Here's the question I REALLY should have asked. Its a little bit more challenging.

Find a power series representation for the function and determine the interval of convergence.

f(x) = x/(2x^2 + 1)

5. Originally Posted by Undefdisfigure
Here's the question I REALLY should have asked. Its a little bit more challenging.

Find a power series representation for the function and determine the interval of convergence.

f(x) = x/(2x^2 + 1)
well, $\frac x{2x^2 + 1} = \frac d{dx} \frac 14 \ln (2x^2 + 1) = \frac 14 \frac d{dx} \ln (1 - (-2x^2))$

and you know the power series for $\ln (1 - x)$, so just find it's derivative and plug it in...

EDIT: in retrospect, it is perhaps easier to think of this as x times the power series of 1/(1 + 2x^2) ...oh well