Question

Find the Taylor series for \[f\left( x \right) = \cos x\].

Answer 1

Hint:
Taylor’s series is an infinite sum of terms that are expressed in terms of the derivatives of the function at a single point.

Complete step by step solution:
The Taylor’s series for a function \[f\left( x \right)\] at \[x = a\](where \[a\] is any real or complex number) can be found by the formula : \[\sum\limits_{n = 0}^\infty {\dfrac{{{f^{\left( n \right)}}\left( a \right)}}{{n!}}} {\left( {x - a} \right)^n}\] (where \[{f^{(n)}}(x)\] is the nth derivative of \[f(x)\] at \[x = a\]).
Let us find the Taylor series for \[f(x) = \cos x\] at \[x = 0\]. First we need to find the derivatives of \[\cos x\] and their values for \[x = 0\].
\[f(x) = \cos x \Rightarrow f(0) = \cos (0) = 1\]
\[f'(x) = - \sin x \Rightarrow f'(0) = - \sin (0) = 0\]

\[f'''(x) = \sin x \Rightarrow f'''(0) = \sin (0) = 0\]

Observe that after \[f'''(x)\], the value of is equal to \[f(x)\] , then \[{f^{(5)}}(x)\] is equal to \[f'(x)\] and the cycle repeats itself after every four values i.e. \[1,0, - 1,0\].
Thus, the sum of the series is as follows:

Substituting the values :
\[ = \cos (0) + \dfrac{{{x^1}}}{{1!}}\left\{ { - \sin (0)} \right\} + \dfrac{{{x^2}}}{{2!}}\left\{ { - \cos (0)} \right\} + \dfrac{{{x^3}}}{{3!}}\left\{ {\sin (0)} \right\} + \dfrac{{{x^4}}}{{4!}}\left\{ {\cos (0)} \right\} + .....\]
\[ = 1 + \dfrac{{{x^1}}}{{1!}}(0) + \dfrac{{{x^2}}}{{2!}}( - 1) + \dfrac{{{x^3}}}{{3!}}(0) + \dfrac{{{x^4}}}{{4!}}(1) + .....\]
\[ = 1 + \dfrac{{{x^2}}}{{2!}}( - 1) + \dfrac{{{x^4}}}{{4!}}(1) - .....\]

It is seen that only the even power of \[x\] exists in the final series and there are alternate positive and negative signs, therefore the summation of the series can be written as:
\[\sum\limits_{n = 0}^\infty {\dfrac{{{{\left( { - 1} \right)}^n}{x^{2n}}}}{{\left( {2n} \right)!}}} \] [ here \[n \in W\] and we take \[2n\] because only the even powers of \[x\] exist and \[{( - 1)^n}\] because of the alternate positive and negative signs.]

Hence, the Taylor’s series for \[f(x) = \cos x\] is:
\[1 - \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^4}}}{{4!}} - \dfrac{{{x^6}}}{{6!}} + ....\]

Note:
The factorial of a number denoted by \[n!\] is equal to the product of all numbers from \[1\] to that number. For example \[4! = 4 \times 3 \times 2 \times 1 = 24\] and \[3! = 3 \times 2 \times 1 = 6\].