Question

Find a power series representation for the following function.

f(x)=ln(1+x). sum_(n=1)^[infinity] ((-1)**n

Answer 1

a. Recall that

For , we have

By integrating both sides, we get

If , then

so that

We can shift the index to simplify the sum slightly.

b. The power series for  can be obtained simply by multiplying both sides of the series above by .

c. We have