Question

How do you find the Maclaurin series of $f\left( x \right) = \cos \left( {{x^2}} \right)$ ?

Answer 1

Hint : A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function. The Maclaurin series of a function f(x) up to order n may be found using Series $\left[ {f,x,0,n} \right]$ and using the formula $f\left( x \right) = \sum\limits_{n = 0}^\infty {\dfrac{{{f^n}\left( {{x_0}} \right)}}{{n!}}} \left( {x - {x_0}} \right)$ we can find the series of the given function.
Formula used:
$f\left( x \right) = \sum\limits_{n = 0}^\infty {\dfrac{{{f^n}\left( {{x_0}} \right)}}{{n!}}} \left( {x - {x_0}} \right)$
$n!$ is factorial of n.
${x_0}$ is a real or complex number.
${f^n}\left( {{x_0}} \right)$ is the nth derivative of f evaluated at the point ${x_0}$ .

Complete step-by-step answer :
The Maclaurin series formula is:
$f\left( x \right) = \sum\limits_{n = 0}^\infty {\dfrac{{{f^n}\left( {{x_0}} \right)}}{{n!}}} \left( {x - {x_0}} \right)$
In which,
$f\left( {{x_0}} \right)$ , $f'\left( {{x_0}} \right)$ , $f''\left( {{x_0}} \right)$ … are the successive differentials when ${x_0} = 0$ .
The Maclaurin series of $\cos x$ is:
$\cos x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\dfrac{{{x^{2n}}}}{{\left( {2n} \right)!}}}$
We need to find for the function, $f\left( x \right) = \cos \left( {{x^2}} \right)$ hence, replace $x$ by ${x^2}$
$\cos \left( {{x^2}} \right) = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\dfrac{{{x^{4n}}}}{{\left( {2n} \right)!}}}$
Therefore, by using $\sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\dfrac{{{x^{4n}}}}{{\left( {2n} \right)!}}}$ we can find the Maclaurin series of the given function $f\left( x \right) = \cos \left( {{x^2}} \right)$ .
So, the correct answer is “ $\sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\dfrac{{{x^{4n}}}}{{\left( {2n} \right)!}}}$ ”.

Note : Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series. A Maclaurin series is a power series that allows one to calculate an approximation of a function $f\left( x \right)$ for input values close to zero, given that one knows the values of the successive derivatives of the function at zero.