# Power Series for Functions in Powers Other than x Examples 1

On the Power Series for Functions in Powers Other than x page, we looked at some examples of obtaining power series representations of functions in powers other than $x$. We will now look at some more examples like that.

## Example 1

**Find a power series representation of $\ln \mid 2 - x \mid$ in powers of $x - 1$. Determine the center of convergence, radius of convergence, and interval of convergence for this power series.**

Let $t = x - 1$. Then $x = t + 1$, and so $\ln \mid 2 - x \mid = \mid 1 - t \mid$. Note that $\ln \mid 1 - t \mid = -\int \frac{1}{1 - t} \: dt$. We have that:

(1)Therefore by integrating this series we get:

(2)Substituting back in $t = x - 1$ and we have that:

(3)The center of convergence is $1$. We must have that $\mid t \mid < 1$ which is equivalent to $\mid x - 1 \mid < 1$ or $0 < x < 2$. Therefore the radius of convergence is also $1$. We now test the endpoints, $0$ and $2$ of our series.

For $x = 0$ we have:

(4)This series converges as an alternating harmonic series. For $x = 2$, we have:

(5)This series diverges as a regular harmonic series. Therefore the interval of convergence is $[0, 2)$.