Here we will show the most typical situations for expanding functions in power series. We will solve them step by step, always all of them simultaneously.
We want to expand the following functions with center 3. The basic function/series is always clear, so we go right to Step 2:
Note that it is not customary to mark the substitution as we did above. We decided to show more detail here since this is a learning place, but normally we usually skip the description and write the outcome of substitution right away.
Step 3: In the exponential case we have to put
together the two series, one truly infinite, one actually finite. This will
be done by separating overlapping terms from the first one.
With the sine and cosine examples we just move terms in front of series into
them; note that the two sums in the cosine example cannot be put together,
since the first only features even powers and the second odd powers.
Actually, this can be written as one series, but the way is quite convoluted
and ugly and people generally prefer the "not exactly proper" form of
answer.
The first three expansions were valid for all y,
therefore also for all x real. In the last case we used the geometric
series where the expansion is true exactly if
