Theorem: Integration of Power Series

27 February 2012, 18:58


This is Rob’s attempt at a snowman, when we went wandering in the Ruahines

Today’s Theorem comes thanks to Simon, because he is stupid. It is given without proof, because I don’t know how to prove it, and my book doesn’t have a proof. I probably don’t even know what it says.

Theorem:
Suppose that a function $f$ is represented by a power series in $x – x_0$ that has a nonzero radius of convergence $R$; that is,
\[
f(x) = \sum_{k=0}^{\infty} c_k(x – x_0)^k \qquad (x_0 < x < x_0 + R).
\]


  1. If the power series representation of $f$ is integrated term by term using an indefinite integral, then the resulting series has radius of converge $R$ and converges to $\int f(x) dx$ on the interval $(x_0 – R, x_0 + R)$; that is,
    \[
    \int f(x)dx = \sum_{k=0}^{\infty} \left[ \int c_k(x – x_0)^k dx \right] + C \qquad (x_0 – R < x < x_0 + R)
    \]
  2. If $\alpha$ and $\beta$ are points in the interval $(x_0 – R, x_0 + R)$, and if the power series representation of $f$ is integrated term by term from $\alpha$ to $\beta$, then the resulting series of numbers converges absolutely on the interval $(x_0 – R, x_0 + R)$ and
    \[
    \int_{\alpha}^{\beta} f(x) dx = \sum_{k=0}^{\infty} \left[\int_{\alpha}^{\beta} c_k(x – x_0)^k dx \right]
    \]

Posted by Michael Welsh at 05:58.
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