MSDS Spring 2018

DATA 605 Fundamental of Computational Mathematics

Jiadi Li

Week 14 Discussion: Taylor Series

Pick any exercise in 8.8 of the calculus textbook. Solve and post your solution. If you have issues doing so, discuss them.

Pg.467 Ex.19 use the Taylor series given in Key Idea 32 to verify the given identity.
\(\frac{d}{dx}(sin\space x)\) = \(cos\space x\)

\(\frac{d}{dx}(sin\space x)\)


= \(\frac{d}{dx}\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}\) = \(\frac{d}{dx}(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ......)\)

= \(\sum_{n=0}^{\infty}(-1)^n\frac{(2n+1)x^{2n}}{(2n+1)!}\) = \(1 - \frac{3x^2}{3!} + \frac{5x^4}{5!} - \frac{7x^6}{7!} + ......\)

= \(\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}\) = \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ......\)

= \(cos\space x\)