# Power Series Expansions

• Function: $$f\left( x \right)$$
Real numbers: $$x$$, $$a$$, $$\xi$$
Remainder term: $${R_n}$$
Whole numbers: $$n$$
1. Taylor series
If a function $$f\left( x \right)$$ has continuous derivatives up to $$\left( {n + 1} \right)$$th order inclusive, then this function can be expanded in a power series about the point $$x = a$$ by the Taylor formula:
$$f\left( x \right) =$$ $$\sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( a \right){\large\frac{{{{\left( {x – a} \right)}^n}}}{{n!}}\normalsize}} =$$ $${f\left( a \right) + f’\left( a \right)\left( {x – a} \right)}$$ $$+\;{{\large\frac{{f^{\prime\prime}\left( a \right){{\left( {x – a} \right)}^2}}}{{2!}}\normalsize} + \ldots}$$ $$+\;{{\large\frac{{{f^{\left( n \right)}}\left( a \right){{\left( {x – a} \right)}^n}}}{{n!}}\normalsize} + {R_n}},$$
where the remainder term $${R_n}$$ in the Lagrange term is given by the expression
$${R_n} = {\large\frac{{{f^{\left( {n + 1} \right)}}\left( \xi \right){{\left( {x – a} \right)}^{n + 1}}}}{{\left( {n + 1} \right)!}}\normalsize},$$ $$a \lt \xi \lt x.$$
If this expansion converges over a certain range of $$x$$ centered at $$a$$, that is, $$\lim\limits_{n \to \infty } {R_n} = 0,$$ then the expansion is called Taylor series of the function $$f\left( x \right)$$ expanded about the point $$a$$.
2. A Maclaurin series is a special case of a Taylor series when the power series expansion is performed at the point $$a = 0:$$
$$f\left( x \right) = \sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( 0 \right){\large\frac{{{x^n}}}{{n!}}}\normalsize} =$$ $$f\left( 0 \right) + f’\left( 0 \right)x$$ $$+\;{\large\frac{{f^{\prime\prime}\left( 0 \right){x^2}}}{{2!}}\normalsize} + \ldots$$ $$+\;{\large\frac{{{f^{\left( n \right)}}\left( 0 \right){x^n}}}{{n!}}\normalsize} + {R_n}.$$

Below are some important Maclaurin series expansions.
3. $${e^x} = 1 + x + {\large\frac{{{x^2}}}{{2!}}\normalsize}$$ $$+\;{\large\frac{{{x^3}}}{{3!}}\normalsize} + \ldots$$ $$+\;{\large\frac{{{x^n}}}{{n!}}\normalsize} + \ldots$$
4. $${a^x} = 1 + {\large\frac{{x\ln a}}{{1!}}\normalsize}$$ $$+\;{\large\frac{{{{\left( {x\ln a} \right)}^2}}}{{2!}}\normalsize} + {\large\frac{{{{\left( {x\ln a} \right)}^3}}}{{3!}}\normalsize} + \ldots$$ $$+\;{\large\frac{{{{\left( {x\ln a} \right)}^n}}}{{n!}}\normalsize} + \ldots$$
5. $$\ln \left( {1 + x} \right) =$$ $$x – {\large\frac{{{x^2}}}{2}\normalsize} + {\large\frac{{{x^3}}}{3}\normalsize}$$ $$-\;{\large\frac{{{x^4}}}{4}\normalsize} + \ldots$$ $$+\;{\large\frac{{{{\left( { – 1} \right)}^n}{x^{n + 1}}}}{{n + 1}}\normalsize} \pm \ldots ,$$ $$– 1 \lt x \le 1$$
6. $$\ln {\large\frac{{1 + x}}{{1 – x}}\normalsize} =$$ $$2\Big( {x + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^5}}}{5}\normalsize} }$$ $${+\;{\large\frac{{{x^7}}}{7}\normalsize} + \ldots } \Big),$$ $$\left| x \right| \lt 1$$
7. $$\ln x =$$ $$2\Big[ {{\large\frac{{x – 1}}{{x + 1}}\normalsize} + {\large\frac{1}{3}\normalsize} {{\left( {\large\frac{{x – 1}}{{x + 1}}\normalsize} \right)}^3} }$$ $$+\;{ {\large\frac{1}{5}\normalsize}{{\left( {\large\frac{{x – 1}}{{x + 1}}\normalsize} \right)}^5} + \ldots } \Big],$$ $$x \gt 0$$
8. $$\cos x =$$ $$1 – {\large\frac{{{x^2}}}{{2!}}\normalsize} + {\large\frac{{{x^4}}}{{4!}}\normalsize} – {\large\frac{{{x^6}}}{{6!}}\normalsize} + \ldots$$ $$+\; {\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n}}}}{{\left( {2n} \right)!}}\normalsize} \pm \ldots$$
9. $$\sin x =$$ $$x – {\large\frac{{{x^3}}}{{3!}}\normalsize} + {\large\frac{{{x^5}}}{{5!}}\normalsize} – {\large\frac{{{x^7}}}{{7!}}\normalsize} + \ldots$$ $$+\; {\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}\normalsize} \pm \ldots$$
10. $$\tan x =$$ $$x + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{2{x^5}}}{{15}}\normalsize}$$ $$+\;{\large\frac{{17{x^7}}}{{315}}\normalsize} + {\large\frac{{62{x^9}}}{{2835}}\normalsize} + \ldots,$$ $$\left| x \right| \lt {\large\frac{\pi }{2}\normalsize}$$
11. $$\cot x =$$ $${\large\frac{1}{x}\normalsize} – \Big( {{\large\frac{x}{3}\normalsize} + {\large\frac{{{x^3}}}{{45}}\normalsize} }$$ $$+\;{ {\large\frac{{2{x^5}}}{{945}}\normalsize} + {\large\frac{{2{x^7}}}{{4725}}\normalsize} + \ldots } \Big),$$ $$\left| x \right| \lt \pi$$
12. $$\arcsin x =$$ $$x + {\large\frac{{{x^3}}}{{2 \cdot 3}}\normalsize} + {\large\frac{{1 \cdot 3{x^5}}}{{2 \cdot 4 \cdot 5}}\normalsize} + \ldots$$ $$+\;{\large\frac{{1 \cdot 3 \cdot 5 \ldots \left( {2n – 1} \right){x^{2n + 1}}}}{{2 \cdot 4 \cdot 6 \ldots \left( {2n} \right)\left( {2n + 1} \right)}}\normalsize} + \ldots,$$ $$\left| x \right| \lt 1$$
13. $$\arccos x =$$ $${\large\frac{\pi }{2}\normalsize} – \Big( {x + {\large\frac{{{x^3}}}{{2 \cdot 3}}\normalsize} + {\large\frac{{1 \cdot 3{x^5}}}{{2 \cdot 4 \cdot 5}}\normalsize} + \ldots}$$ $${+\;{\large\frac{{1 \cdot 3 \cdot 5 \ldots \left( {2n – 1} \right){x^{2n + 1}}}}{{2 \cdot 4 \cdot 6 \ldots \left( {2n} \right)\left( {2n + 1} \right)}}\normalsize} + \ldots} \Big),$$ $$\left| x \right| \lt 1$$
14. $$\arctan x =$$ $$x – {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^5}}}{5}\normalsize} – {\large\frac{{{x^7}}}{7}\normalsize} + \ldots$$ $$+\;{\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n + 1}}}}{{2n + 1}}\normalsize} \pm \ldots ,$$ $$\left| x \right| \le 1$$
15. $$\cosh x =$$ $$1 + {\large\frac{{{x^2}}}{{2!}}\normalsize} + {\large\frac{{{x^4}}}{{4!}}\normalsize} + {\large\frac{{{x^6}}}{{6!}}\normalsize} + \ldots$$ $$+\;{\large\frac{{{x^{2n}}}}{{\left( {2n} \right)!}}\normalsize} + \ldots$$
16. $$\sinh x =$$ $$x + {\large\frac{{{x^3}}}{{3!}}\normalsize} + {\large\frac{{{x^5}}}{{5!}}\normalsize} + {\large\frac{{{x^7}}}{{7!}}\normalsize} + \ldots$$ $$+\;{\large\frac{{{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}\normalsize} + \ldots$$