Power Series
Earlier, we found out that old friends like sine, cosine, and the exponential function live secret lives as infinite sums. In fact,
using differential equations. It's natural to ask, “Given a function, how do we find its sum representation?” and “What are these sums good for?” We'll attempt to answer both these questions in this quiz.
First, some terminology. If
we call the series the Taylor expansion of centered at
, where the center
is fixed.
For convenience, we'll choose the center to be 0 always, but this doesn't have to be the case. The constants are called the coefficients.
There is a formula for the coefficients that we develop in the next problem.
Let's assume ; we want to deduce the values of the coefficients using properties of f(x).f(x). To start, we see that
,
so . If we take the derivative of the power series and plug x = 0, the only term that survives is
Therefore, . Continue this approach for a few more coefficients and come up with a formula for
.
不做了，简书的数学公式输入实在是太不方便了！！！
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