# Taylors Series Confusion

• Apr 28th 2011, 05:56 AM
Redtwit
Taylors Series Confusion
Hello, just thought i'd quickly say, I'm new here =].

I'm doing some problems on Taylors Series, like this one:

Find the Taylor polynomial of order 2 at x = a generated by the
function f(x) = sin(sin x) at a = 0:

I did this, coming to the correct answer by finding the 2 derivatives etc. The solutions however gave me this, which was a lot difference from my working:

sin(sin x) = sin x + O(sin^3 x) = x + O(x^3);
Hence P2(x) = x:

I was hoping somebody could perhaps explain the O symbol to me? and how sin(sin x) = sin c+ O(sin^3 x) etc.

Another is:

Verify that (sin x)/x = 1 - (x^2)/6 + o(x^3)

Thanks =]
• Apr 28th 2011, 11:01 PM
chisigma
Quote:

Originally Posted by Redtwit
Hello, just thought i'd quickly say, I'm new here =].

I'm doing some problems on Taylors Series, like this one:

Find the Taylor polynomial of order 2 at x = a generated by the
function f(x) = sin(sin x) at a = 0:

I did this, coming to the correct answer by finding the 2 derivatives etc. The solutions however gave me this, which was a lot difference from my working:

sin(sin x) = sin x + O(sin^3 x) = x + O(x^3);
Hence P2(x) = x:

I was hoping somebody could perhaps explain the O symbol to me? and how sin(sin x) = sin c+ O(sin^3 x) etc.

Use the indentity...

... and then find the McLaurin expansion of each exponential in (1) proceeding as in...

http://www.mathhelpforum.com/math-he...nz-178887.html

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Apr 28th 2011, 11:14 PM
chisigma
Quote:

Originally Posted by Redtwit
Hello, just thought i'd quickly say, I'm new here =].

Verify that (sin x)/x = 1 - (x^2)/6 + o(x^3)

Starting from the well known...

http://quicklatex.com/cache3/ql_4941...83ecf81_l3.png (1)

... deviding both terms by x You obtain...

http://quicklatex.com/cache3/ql_dddc...14ea24c_l3.png (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$