While tidying up my old University Maths notes the other day I came across this problem. (I didn't get it right first time round!)
It's quite interesting and I thought I'd also use the opportunity to setup nice looking maths on the blog. So, the question:
Find the first two terms of the series for tanx in powers of x.
Now,
tanx=\frac{sinx}{cosx}
and the power series expansions of sinx and cosx begin as:
sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...
cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...
Put this together:
tanx=\frac{x-\frac{x^3}{6}+...}{1-\frac{x^2}{2}+...}
Multiply by \frac{1+\frac{x^2}{2}}{1+\frac{x^2}{2}} (This is just multiplying by 1 and chosen so we can deal with the x^2 term in the denominator), so:
tanx=\frac{(1+\frac{x^2}{2})(x-\frac{x^3}{6}+...)}{(1+\frac{x^2}{2})(1-\frac{x^2}{2}+...)}
If we now just multiply out the first term we get
tanx=\frac{x+\frac{x^3}{2}-\frac{x^3}{6}+...}{1+\frac{x^2}{2}-\frac{x^2}{2}+...}
tanx=\frac{x+\frac{x^3}{3}+...}{1+...}
The other terms in x^4 and x^5 can be ignored, giving:
tanx=x+\frac{x^3}{3}+...
To display the formulae I've added MathJax to the blog.
Go to Theme then from the Customise dropdown select Edit HTML. Add the following just below the <head> element:
| <script src='http://cdn.mathjax.org/mathjax/latest/MathJax.js' type='text/javascript'> | |
| MathJax.Hub.Config({ | |
| extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"], | |
| jax: ["input/TeX", "output/HTML-CSS"], | |
| tex2jax: { | |
| inlineMath: [ ['',''], ["\\(","\\)"] ], | |
| displayMath: [ ['',' '], ["\\[","\\]"] ], |
|
| }, | |
| "HTML-CSS": { availableFonts: ["TeX"] } | |
| }); | |
| </script> |
Finally, here's a useful link: MathJax basic tutorial and quick reference - Mathematics Meta Stack Exchange
Oops, missed the first x when multiplying out the numerator in the first version of this post. Now updated!
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