How do you find the power series for $f(x)=1/(1+3x)$ and determine its radius of convergence?

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Answer 1 · 2017-01-09T17:24:04

$1/(1+3x) = sum_(n=0)^oo (-3x)^n$ for $abs(x) < 1/3$

We know that the geometric series:

$sum_(n=0)^oo t^n$

converges for $abs(t) < 1$ and the sum is:

$sum_(n=0)^oo t^n = 1/(1-t)$

We can then substitute:

$t=-3x$

and see that:

$1/(1+3x) = sum_(n=0)^oo (-3x)^n$

for

$abs (-3x) < 1$ , or $abs(x) < 1/3$

How do you find the power series for f(x)=1/(1+3x)f(x)=1/(1+3x) and determine its radius of convergence?