# How do you find a power series converging to #f(x)=int ln(1+t^2) dt# from [0,x] and determine the radius of convergence?

##### 2 Answers

Substitute the series for

#### Explanation:

The Taylor series for

Next, replace

Now integrate this term-by-term, from

The original radius of convergence for

The interval of convergence does get "expanded" by one point, however. The interval of convergence for the final answer is

for

#### Explanation:

Note that:

Now:

can be expressed as the sum of a geometric series of ratio

and then:

Inside the interval of convergence we can integrate term by term to have a power series with the same radius of convergence:

and integrating again: