Question

Find a power series representation for the function. (give your power series representation centered at x = 0. ) f(x) = ln(5 − x)

Answer 1

Recall that for $|x|$, we have the convergent geometric series

$\displaystyle \sum_{n=0}^\infty x^n = \frac1{1-x}$

Now, for $\left|\frac x5\right| < 1$, we have

$\dfrac1{5 - x} = \dfrac15 \cdot \dfrac1{1 - \frac x5} = \dfrac15 \displaystyle \sum_{n=0}^\infty \left(\frac x5\right)^n = \sum_{n=0}^\infty \frac{x^n}{5^{n+1}}$

Integrating both sides gives

$\displaystyle \int \frac{dx}{5-x} = C + \int \sum_{n=0}^\infty \frac{x^n}{5^{n+1}} \, dx$

$\displaystyle -\ln(5-x) = C + \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}$

If we let $x=0$, the sum on the right side drops out and we're left with $C=-\ln(5)$.

It follows that

$\displaystyle \ln(5-x) = \ln(5) - \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}$

or

$\displaystyle \ln(5-x) = \boxed{\ln(5) - \sum_{n=1}^\infty \frac{x^n}{5^n n}}$