New series from known ones

Three quick questions from earlier lessons.

To get the series of $x\sin x$ , can you just multiply the series of $\sin x$ by $x$ ?

Build new series from known ones.

Inside its interval, a power series can be multiplied by

x^k

, have

x

replaced by

u(x)

(such as

-x^2

or

3x

), and be differentiated or integrated term by term. The radius stays the same for differentiation and integration.

The series of $\ln(1 + x)$ .

\dfrac{1}{1 + x} = 1 - x + x^2 - x^3 + \dotsb

Substitute −x into 1/(1 − u).

\ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dotsb

Integrate term by term from 0.

Your turn: the coefficient of $x^3$ in $x^2 e^x$ .

1

x² · x¹/1!.

Find the coefficient of $x^{ 10 }$ in the Maclaurin series of $\dfrac{1}{1 + x^2}$ .
Find the coefficient of $x^{ 2 }$ in the Maclaurin series of $\dfrac{1}{1 + x^2}$ .

$\arctan x = \displaystyle\int_0^x \dfrac{dt}{1 + t^2}$ . Find the coefficient of $x^{ 3 }$ in its Maclaurin series.
$\arctan x = \displaystyle\int_0^x \dfrac{dt}{1 + t^2}$ . Find the coefficient of $x^{ 9 }$ in its Maclaurin series.

Three questions to check you have got it.

New series from old

Watch out: integrating divides by the new power.

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