Question

How do you find a power series representation for `f(x)= 1/(3-x)` and what is the radius of convergence?

Answer 1

The radius of convergence is -3< x <3

Explanation:

`f(x)= 1/(3-x) = 1/(3(1-x/3))`

Now compare this epression with the sum of an infinite geometric series 1 +x+x^2 +x^3 +...... . The sum is `1/(1-x)` with -1< x <1.

Now substitute x with `x/3`, then `1/(1-x/3) = 1 +x/3 +(x/3)^2 +(x/3)^3 +........` where `-1< x/3 <1 -> -3< x < 3`

Hence `f(x)= 1/(3-x)= 1/3 (1 +(x/3) + (x/3)^2 +........)`

The radius of convergence is therefore -3< x <3