(PS6) Maclaurin Series#
In this lesson we are going to see how to:
use the Maclaurin Series for a given function to find the Maclaurin Series expansion of a related function.
Review Videos#
Example 1#
Use the Maclaurin Series for \(\sin x\) to find the Maclaurin Series for \(\cos x\). Find the Radius of Convergence of each series.
\[
\sin x = \sum_{n=0}^{\infty}(-1)^n \dfrac{x^{2n+1}}{(2n+1)!}
\]
Example 2#
Use the Maclaurin Series for \(\cos x\) to find the Maclaurin Series for the following function.
\[
f(x)=x^3\cos 2x
\]
Example 3#
Use the Maclaurin Series for \(e^x\) to find the Maclaurin Series for \(e^{-x^2}\). Find the Radius of Convergence as well.
\[
e^x = \sum_{n=0}^{\infty}\dfrac{x^n}{n!}
\]
Example 4#
Use the Maclaurin Series for \(e^{-x^2}\) to calculate the following integral:
\[
\int_0^{1} e^{-x^2}\, dx
\]
Important Maclaurin Series#
Important Maclaurin Series
\[\begin{split}
&e^x = \sum_{n=0}^{\infty}\dfrac{x^n}{n!} & = 1+\dfrac{x}{1!}+\dfrac{x^2}{2!}+\dfrac{x^3}{3!} + \cdots \\[10pt]
&\sin x = \sum_{n=0}^{\infty}(-1)^n \dfrac{x^{2n+1}}{(2n+1)!} \quad & = x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!} + \cdots \\[10pt]
&\cos x = \sum_{n=0}^{\infty}(-1)^n \dfrac{x^{2n}}{(2n)!} & = 1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!} + \cdots \\[10pt]
\end{split}\]
(More) Important Maclaurin Series
\[\begin{split}
&\dfrac{1}{1-x} = \sum_{n=0}^{\infty}x^n & = 1+x+x^2+x^3 + \cdots \\[10pt]
&\tan^{-1} x =\sum_{n=0}^{\infty}(-1)^n \dfrac{x^{2n+1}}{2n+1} \quad & = x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\dfrac{x^7}{7} + \cdots \\[10pt]
& \ln(1+x) = \sum_{n=1}^{\infty}(-1)^{n-1} \dfrac{x^n}{n} & = x-\dfrac{x^2}{2}+\dfrac{x^3}{3} - \dfrac{x^4}{4}+ \cdots \\[10pt]
\end{split}\]