"Use power series operations to find the Taylor..." solved and explained

pancha3

Answered question

2020-11-29

Use power series operations to find the Taylor series at x=0 for the following function.

\frac{5 x^{2}}{2} - 5 + 5 \cos x

Answer & Explanation

Alara Mccarthy

Skilled2020-11-30Added 85 answers

Two Taylor series at x=0 can be added term by term, giving rise to another Taylor series. The radius of convergence is going to be the minimum of the radii of convergence of the two (Taylor) power series. In our case, the term

(\frac{5 x^{2}}{2} - 5)

is already a Taylor series at x=0. The Taylor series for cosxcosx at x=0 is given by

1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots + {(- 1)}^{n} \frac{x^{2} n}{(2 n)!} + \dots

Adding term by term the two Taylor series, we get:

(- 5 + 1) + (\frac{5}{2} - \frac{1}{2}) x^{2} + \frac{x^{4}}{4!} = \frac{x^{6}}{6!} + \dots + {(- 1)}^{n} \frac{x^{2} n}{(2 n)!} + \dots

which is equal to

- 4 + 2 x^{2} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots + {(- 1)}^{n} \frac{x^{2} n}{(2 n)!} + \dots

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