# Differentiating and Integrating Power Series

## Key Questions

• If $f \left(x\right) = {\sum}_{k = 0}^{\infty} {c}_{k} {x}^{k}$, then
${f}^{\left(n\right)} \left(x\right) = {\sum}_{k = n}^{\infty} k \left(k - 1\right) \left(k - 2\right) \cdots \left(k - n + 1\right) {c}_{k} {x}^{k - n}$

By taking the derivative term by term,
$f ' \left(x\right) = {\sum}_{k = 1}^{\infty} k {c}_{k} {x}^{k - 1}$
$f ' ' \left(x\right) = {\sum}_{k = 2}^{\infty} k \left(k - 1\right) {c}_{k} {x}^{k - 2}$
$f ' ' ' \left(x\right) = {\sum}_{k = 3}^{\infty} k \left(k - 1\right) \left(k - 2\right) {c}_{k} {x}^{k - 3}$
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${f}^{\left(n\right)} \left(x\right) = {\sum}_{k = n}^{\infty} k \left(k - 1\right) \left(k - 2\right) \cdots \left(k - n + 1\right) {c}_{k} {x}^{k - n}$

• If ${\sum}_{n = 0}^{\infty} {c}_{n} {x}^{n}$ is a power series, then its general antiderivative is

$\int {\sum}_{n = 0}^{\infty} {c}_{n} {x}^{n} \mathrm{dx} = {\sum}_{n = 0}^{\infty} {c}_{n} / \left\{n + 1\right\} {x}^{n + 1} + C$.

(Note that integration can be done term by term.)

I hope that this was helpful.

• One of the most useful properties of power series is that we can take the derivative term by term. If the power series is

$f \left(x\right) = {\sum}_{n = 0}^{\infty} {c}_{n} {x}^{n}$,

then by applying Power Rule to each term,

$f ' \left(x\right) = {\sum}_{n = 0}^{\infty} {c}_{n} n {x}^{n - 1} = {\sum}_{n = 1}^{\infty} n {c}_{n} {x}^{n - 1}$.

(Note: When $n = 0$, the term is zero.)

I hope that this was helpful.