- #1
ognik
- 643
- 2
I got a series which was accurate for x=0, but less so for x= $ \frac{\pi}{4}\: (2.422\: instead\: of \: 2.718) $ and decreasing accuracy for $ \frac{\pi}{2} < x < \frac{\pi}{4} $
I used 1st 3 terms of $ e^{x} = 1 + x + \frac{{x}^{2}}{2!} + \frac{{x}^{3}}{3!} + $
$ \therefore e^{Tan(x)} = 1 +Tan(x) + \frac{{\left(Tan(x)\right)}^{2}}{2!} $
I used Tan(x) (from a previous exercise $ = x + \frac{{x}^{3}}{3} + \frac{2{x}^{2}}{15} + $
and finally, discarding terms > x5:
$ e^{Tan(x)}= 1 + x + \frac{1}{2}{x}^{2} + \frac{{x}^{3}}{3} + \frac{{x}^{4}}{3} + \frac{2{x}^{2}}{15} $
Am I close? (I 'feel' like the expansion should be accurate to the 1st decimal at least). Is there a better way to do the expansion (using Taylor though)?
Thanks for reading.
I used 1st 3 terms of $ e^{x} = 1 + x + \frac{{x}^{2}}{2!} + \frac{{x}^{3}}{3!} + $
$ \therefore e^{Tan(x)} = 1 +Tan(x) + \frac{{\left(Tan(x)\right)}^{2}}{2!} $
I used Tan(x) (from a previous exercise $ = x + \frac{{x}^{3}}{3} + \frac{2{x}^{2}}{15} + $
and finally, discarding terms > x5:
$ e^{Tan(x)}= 1 + x + \frac{1}{2}{x}^{2} + \frac{{x}^{3}}{3} + \frac{{x}^{4}}{3} + \frac{2{x}^{2}}{15} $
Am I close? (I 'feel' like the expansion should be accurate to the 1st decimal at least). Is there a better way to do the expansion (using Taylor though)?
Thanks for reading.