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|<1`, 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

`\frac1{5 - x} = \frac15 \cdot \frac1{1 - \frac x5} = \frac15 \displaystyle \sum_(n=0)^\infty \left(\frac x5\right)^n = \sum_(n=0)^\infty (x^n)/(5^(n+1))`

Integrating both sides gives

`\displaystyle \int (dx)/(5-x) = C + \int \sum_(n=0)^\infty (x^n)/(5^(n+1)) \, dx`

`\displaystyle -\ln(5-x) = C + \sum_(n=0)^\infty (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 (x^(n+1))/(5^(n+1)(n+1))`

or

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