Power series expansions
$$ 1. \ e^x = 1+x+ \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots + \frac {x^n}{n!}+ \cdots $$
$$ 2. \ a^x=1+ \frac {x \ln a}{1!}+ \frac {(x \ln a)^2}{2!} + \frac {(x \ln a)^3}{3!}+ \cdots +\frac {(x \ln a)^n}{n!}+\cdots $$
$$ 3. \ \ln (1+x)=x- \frac {x^2}{2} + \frac {x^3}{3} – \frac {x^4}{4}+\cdots + \frac {(-1)^nx^{n+1}}{n+1} \pm \cdots , -1 \lt x \le 1 $$
$$ 4. \ \ln \frac {1+x}{1-x} = 2 \left( x+ \frac {x^3}{3} + \frac {x^5}{5} + \frac {x^7}{7} + \cdots \right), |x| \lt 1 $$
$$ 5. \ \ln x = 2 \left[ \frac {x-1}{x+1} + \frac 13 \left( \frac {x-1}{x+1} \right)^3 + \frac 15 \left( \frac {x-1}{x+1} \right)^5+ \cdots \right] $$
$$ 6. \ \cos x = 1- \frac {x^2}{2!} + \frac {x^4}{4!} – \frac {x^6}{6!} +\cdots+ \frac {(-1)^nx^{2n}}{(2n)!} \pm \cdots $$
$$ 7. \ \sin x = x- \frac {x^3}{3!} + \frac {x^5}{5!} – \frac {x^7}{7!} +\cdots+ \frac {(-1)^nx^{2n+1}}{(2n+1)!} \pm \cdots $$
$$ 8. \ \tan x = x+ \frac {x^3}{3} + \frac {2x^5}{15} + \frac {17x^7}{315} +\cdots, |x| \lt \frac {\pi}{2} $$
$$ 9. \ \cot x = \frac 1x – \left( \frac x3 + \frac {x^3}{45} + \frac {2x^5}{945} + \frac {2x^7}{4725} +\cdots \right), |x| \lt \pi $$
$$ 10. \ \sin^{-1}x = x+ \frac {x^3}{2\cdot 3} + \frac {1\cdot 3 x^5}{2\cdot 4 \cdot5} + \cdots + \frac {1\cdot 3\cdot 5 \cdots (2n-1) x^{2n+1}}{2\cdot 4 \cdot6 \cdots (2n)(2n+1)} + \cdots , \ |x| \lt 1 $$
$$ 11. \ \cos^{-1}x = \frac {\pi}{2} – \left( x+ \frac {x^3}{2\cdot 3} + \frac {1\cdot 3 x^5}{2\cdot 4 \cdot5} + \cdots + \frac {1\cdot 3\cdot 5 \cdots (2n-1) x^{2n+1}}{2\cdot 4 \cdot6 \cdots (2n)(2n+1)} + \cdots \right) , \ |x| \lt 1 $$
$$ 12. \ \tan^{-1}x = x- \frac {x^3}{3} + \frac {x^5}{5} – \frac {x^7}{7} + \dots + \frac {(-1)^n x^{2n+1}}{2n+1} \pm \cdots, |x| \le 1 $$
$$ 13. \ \cosh x = 1+ \frac {x^2}{2!} + \frac {x^4}{4!} + \frac {x^6}{6!} + \cdots + \frac {x^{2n}}{(2n)!} + \cdots $$
$$ 14. \ \sinh x = x+ \frac {x^3}{3!} + \frac {x^5}{5!} + \frac {x^7}{7!} + \cdots + \frac {x^{2n+1}}{(2n+1)!} + \cdots $$
Example:
$$ \text{Find the series for} \ (x+1) \sin x $$
Solution:
$$ (x+1) \sin x = (x+1) \left( x- \frac {x^3}{3!} – \frac {x^5}{5!}- \right)\cdots $$
$$ (x+1) \sin x = x+x^2- \frac {x^3}{3!} – \frac {x^4}{3!}+ \cdots $$
Example:
$$ \text{Find the series for} \ e^x \cos x $$
Solution:
$$ e^x \cos x = \left( 1+x+ \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots \right) \left( 1- \frac {x^2}{2!} + \frac {x^4}{4!}- \cdots \right) $$
$$ e^x \cos x = 1+x+ \frac {x^2}{2!} + \frac {x^3}{3!}+ \cdots – \frac {x^2}{2!} – \frac {x^3}{2!} – \frac {x^4}{2!2!} + \cdots $$
$$ e^x \cos x = 1+x+0x^2- \frac {x^3}{3} – \frac {x^4}{6} + \cdots $$
$$ e^x \cos x = 1+x-\frac {x^3}{3} – \frac {x^4}{6} + \cdots $$