For what values of p does the Taylor series for \(f(x)=(1+x)^{p}\) centered at 0 terminate?

Solution 6E

Step 1:

In this problem we have to find for what values of p does the Taylor series for centered at 0 terminate.

Taylor series centered at 0 is given by,

We have

When p=0,

The corresponding Taylor series is 1 (which terminates)

When p=1,

The corresponding Taylor series is (which terminates)

When p=2,

The corresponding Taylor series is (which terminates)

When p= -1,

The corresponding Taylor series is

Therefore by above observations we conclude that for positive values of p the Taylor series for centered at 0 terminate. That is when .