Find a power series representation for the function. Determine the radius of convergence, R$.($Give your power series representation centered at x $=0.)$

f$($x$)=$ ln$(10$ x$)$

Step $1$

We must first recognize that

f$($x$)=$ ln$(10$ x$)$

is an anti$-$derivative of a more familiar function. To find this function, we find

ddx

ln$(10$ x$)$

$=$

$$$1$x$10$

$.$

Step $2$

Since

ddx

ln$(10$ x$)$

$=$

$110$ x

$,$

$$

$110$ x

$$dx

$=$

$.$

$($Remember to use absolute values where appropriate. Use C for the constant of integration.$)$