Finding Taylor series about some point

[JG89]

In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf


It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0.

It says we have to find the Taylor series for sinx about x = 1, and then we can substitute 2x + 1 into it.

Why is this?

 

[mathman]

The Taylor series for sinu around 1 will be a series of powers of (u-1). Substitute 2x+1 for u and you will have a series in powers of 2x.