Proving the Taylor Series of Sin(x)

**Leaf** · November 21st 2010, 07:38 AM

Could someone help me with this problem? Thanks.

"Prove that the Taylor series of sin(x) at a=pi/2 represents sin(x) for all x."

I already know how to find the Taylor series, but I have no idea how to prove anything.

**TheEmptySet** · November 21st 2010, 08:46 AM

Originally Posted by Leaf

Could someone help me with this problem? Thanks.

"Prove that the Taylor series of sin(x) at a=pi/2 represents sin(x) for all x."

I already know how to find the Taylor series, but I have no idea how to prove anything.

Use Taylors' remainder theorem

Taylor's theorem - Wikipedia, the free encyclopedia

Note that for all $x \in \mathbb{R}$

$|\sin(x)|\le 1$ and $|\cos(x)|\le 1$ . Also that for each x that the remainder (the error) goes to $0 \text{ as } n \to \infty$

**Leaf** · November 21st 2010, 05:01 PM

I don't think it works. The Taylor Series only represents sin(x) when x is between (pi/2,pi/2] when the problem asks for all x. I can't prove something that's not true.

**TheEmptySet** · November 21st 2010, 07:03 PM

Originally Posted by Leaf

I don't think it works. The Taylor Series only represents sin(x) when x is between (pi/2,pi/2] when the problem asks for all x. I can't prove something that's not true.

I think you are mistaken. Both $\sin(x)$ and $\cos(x)$ are real analytic entire functions.

Consider the remainder term from the link above it gives

$\displaystyle R_n(x)=\frac{f^{n+1}(c)}{(n+1)!}\left(x-\frac{\pi}{2} \right)^{n+1}$

Since all derivatives of $\sin(x)$ are less than 1

$\displaystyle |R_n(x)| \le \frac{1}{(n+1)!}\left|x-\frac{\pi}{2} \right|^{n+1}$

now for any $y \in \mathbb{R}$ we can choose an $n$ so large that the remainder is as small as we wish

$\displaystyle |R_n(y)| \le \frac{1}{(n+1)!}\left|y-\frac{\pi}{2} \right|^{n+1} < \epsilon$

**Leaf** · November 22nd 2010, 03:31 AM

Alright, thanks for the help. I think I got it now.

Thread: Proving the Taylor Series of Sin(x)

LinkBack

Thread Tools

Display

Proving the Taylor Series of Sin(x)

Similar Math Help Forum Discussions

question about power series, taylor series, maclaurin series

Taylor Expansion, and proving the second derivative??

Proving a taylor series converges.

Proving remainder in Taylor series grpahically.

Using elementary taylor series for taylor polynomial

Search Tags