Standard Series and Validity

TGM Original Questions

The following standard series are stated in MF27. Their validity conditions are part of the result and must be carried into any substituted or combined expansion.

Standard series

Function	Expansion	Validity
$(1+x)^n$	$1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots+\dfrac{n(n-1)\cdots(n-r+1)}{r!}x^r+\cdots$	General real $n$: $\lvert x\rvert<1$. Integer $n\geq0$: terminates; all real $x$.
$e^x$	$1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^r}{r!}+\cdots$	All real $x$
$\sin x$	$x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots+\dfrac{(-1)^r x^{2r+1}}{(2r+1)!}+\cdots$	All real $x$
$\cos x$	$1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots+\dfrac{(-1)^r x^{2r}}{(2r)!}+\cdots$	All real $x$
$\ln(1+x)$	$x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots+\dfrac{(-1)^{r+1}x^r}{r}+\cdots$	$-1<x\leq1$

Validity after substitution

When a standard series is used with an inner expression, apply its condition to that whole expression. For example, $\ln(1-3x)$ requires $-1<-3x\leq1$, so $-\dfrac13\leq x<\dfrac13$. For a product or combination of series, use the intersection of all relevant validity intervals.

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Answer:$\displaystyle f(x)=\sum_{r=0}^{\infty}\frac{f^{(r)}(0)}{r!}x^r$; $\displaystyle (1+x)^n=\sum_{r=0}^{\infty}\binom{n}{r}x^r$ for $|x|<1$; $\displaystyle e^x=\sum_{r=0}^{\infty}\frac{x^r}{r!}$; $\displaystyle \sin x=\sum_{r=0}^{\infty}\frac{(-1)^r x^{2r+1}}{(2r+1)!}$; $\displaystyle \cos x=\sum_{r=0}^{\infty}\frac{(-1)^r x^{2r}}{(2r)!}$; $\displaystyle \ln(1+x)=\sum_{r=1}^{\infty}\frac{(-1)^{r+1}x^r}{r}$ for $-1<x\le1$

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Function	Expansion	Validity
\((1+x)^n\)	\(1+nx+\dfrac{n(n-1)}{2!}x^2+\cdots+\dfrac{n(n-1)\cdots(n-r+1)}{r!}x^r+\cdots\)	General real \(n\): \(\lvert x\rvert<1\). Integer \(n\geq0\): terminates; all real \(x\).
\(e^x\)	\(1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^r}{r!}+\cdots\)	All real \(x\)
\(\sin x\)	\(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots+\dfrac{(-1)^r x^{2r+1}}{(2r+1)!}+\cdots\)	All real \(x\)
\(\cos x\)	\(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots+\dfrac{(-1)^r x^{2r}}{(2r)!}+\cdots\)	All real \(x\)
\(\ln(1+x)\)	\(x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots+\dfrac{(-1)^{r+1}x^r}{r}+\cdots\)	\(-1<x\leq1\)