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# 12.3 Differentiation of Power Series

If is a power series, then the series obtained by differentiating the terms of is

This is not a power series, but its translate

is.

12.15   Definition (Formal derivative.) If is a power series, then the formal derivative of is

I will sometimes write when I think this will cause no confusion.

12.16   Examples.

or

Our fundamental theorem on power series is:

12.17   Theorem (Differentiation theorem.) Let be a power series. Then and have the same radius of convergence. The function associated with is differentiable in the disc of convergence, and the function represented by agrees with on the disc of convergence.

The proof is rather technical, and I will postpone it until section 12.8. I will derive some consequences of it before proving it (to convince you that it is worth proving).

12.18   Example. We know that the geometric series has radius of convergence and for . The differentiation theorem says also has radius of convergence , and

i.e.,

We can apply the theorem again and get

or

Another differentiation gives us

or

The pattern is clear, and I omit the induction proof that for all

12.19   Exercise. By assuming the differentiation theorem, we've shown that the series has radius of convergence for all . Verify this directly.

12.20   Exercise. A Find formulas for and that are valid for . (You may assume the differentiation theorem.)

12.21   Example. By the differentiation theorem, if

then and are differentiable on and , and . (We saw in earlier examples that both series have radius of convergence , and that the formal derivatives satisfy and .) Also, clearly and are real when is real. The discussion in example 10.3 then shows that for real , and agree with the cosine and sine functions you discussed in your previous calculus course, and in particular that

Next: 12.4 The Exponential Function Up: 12. Power Series Previous: 12.2 Radius of Convergence   Index