# Using Maclauriw series to find first 3 terms

• Nov 4th 2012, 12:19 AM
Using Maclauriw series to find first 3 terms
Hello everyone, i need some help on this question :
By using the Maclauriw series, show that the first three terms of the expansion ln[sec(2x)+tan(2x)] in powers of x are 2x + [4(x^3)]/3 + [4(x^5)]/3.
Thanks! ;)
• Nov 4th 2012, 01:57 AM
Prove It
Re: Using Maclauriw series to find first 3 terms
It will be easiest if you find the MacLaurin Series for \displaystyle \displaystyle \begin{align*} \sec{(2x)} \end{align*} and integrate it, since \displaystyle \displaystyle \begin{align*} \int{\sec{(2x)}\,dx} = \ln{\left[ \sec{(2x)} + \tan{(2x)} \right]} \end{align*}.
• Nov 4th 2012, 03:45 AM
HallsofIvy
Re: Using Maclauriw series to find first 3 terms
Quote:

Originally Posted by Difficultofindtheusername
Hello everyone, i need some help on this question :
By using the Maclauriw series, show that the first three terms of the expansion ln[sec(2x)+tan(2x)] in powers of x are 2x + [4(x^3)]/3 + [4(x^5)]/3.
Thanks! ;)

Are you saying that you do not know what the MacLaurin series is? The MacLaurin series for function f is the Taylor's series about x= 0. It's first three terms are f(0)+ f'(0)x+ f''(0)x^2/2. So the first thing you need to do is find the first and second derivatives of ln[sec(2x)+ tan(2x)] and evaluate at x= 0. Tedious, but nothing deep.
• Nov 5th 2012, 05:30 AM
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