Important Formulas

Differentiation Rules

1. $\dfrac{d}{dx}(cx)=c$	10. $\dfrac{d}{dx}(a^{x})=\ln a\cdot a^{x}$	19. $\dfrac{d}{dx}(\sin^{-1}x)=\dfrac{1}{\sqrt{1-x^{2}}}$	28. $\dfrac{d}{dx}(\operatorname{sech}x)=-\operatorname{sech}x\tanh x$
2. $\dfrac{d}{dx}(u\pm v)=u^{\prime}\pm v^{\prime}$	11. $\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$	20. $\dfrac{d}{dx}(\cos^{-1}x)=\frac{-1}{\sqrt{1-x^{2}}}$	29. $\dfrac{d}{dx}(\operatorname{csch}x)=-\operatorname{csch}x\coth x$
3. $\dfrac{d}{dx}(u\cdot v)=uv^{\prime}+u^{\prime}v$	12. $\dfrac{d}{dx}(\log_{a}x)=\dfrac{1}{x\ln a}$	21. $\dfrac{d}{dx}(\csc^{-1}x)=\frac{-1}{\left\lvert x\right\rvert\sqrt{x^{2}-1}}$	30. $\dfrac{d}{dx}(\coth x)=-\operatorname{csch}^{2}x$
4. $\dfrac{d}{dx}(\dfrac{u}{v})=\dfrac{vu^{\prime}-uv^{\prime}}{v^{2}}$	13. $\dfrac{d}{dx}(\sin x)=\cos x$	22. $\dfrac{d}{dx}(\sec^{-1}x)=\dfrac{1}{\left\lvert x\right\rvert\sqrt{x^{2}-1}}$	31. $\dfrac{d}{dx}(\cosh^{-1}x)=\frac{1}{\sqrt{x^{2}-1}}$
5. $\dfrac{d}{dx}(u(v))=u^{\prime}(v)v^{\prime}$	14. $\dfrac{d}{dx}(\cos x)=-\sin x$	23. $\dfrac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^{2}}$	32. $\dfrac{d}{dx}(\sinh^{-1}x)=\frac{1}{\sqrt{x^{2}+1}}$
6. $\dfrac{d}{dx}(c)=0$	15. $\dfrac{d}{dx}(\csc x)=-\csc x\cot x$	24. $\dfrac{d}{dx}(\cot^{-1}x)=\frac{-1}{1+x^{2}}$	33. $\dfrac{d}{dx}(\operatorname{sech}^{-1}x)=\frac{-1}{x\sqrt{1-x^{2}}}$
7. $\dfrac{d}{dx}(x)=1$	16. $\dfrac{d}{dx}(\sec x)=\sec x\tan x$	25. $\dfrac{d}{dx}(\cosh x)=\sinh x$	34. $\dfrac{d}{dx}(\operatorname{csch}^{-1}x)=\frac{-1}{\left\lvert x\right\rvert% \sqrt{1+x^{2}}}$
8. $\dfrac{d}{dx}(x^{n})=nx^{n-1}$	17. $\dfrac{d}{dx}(\tan x)=\sec^{2}x$	26. $\dfrac{d}{dx}(\sinh x)=\cosh x$	35. $\dfrac{d}{dx}(\tanh^{-1}x)=\frac{1}{1-x^{2}}$
9. $\dfrac{d}{dx}(e^{x})=e^{x}$	18. $\dfrac{d}{dx}(\cot x)=-\csc^{2}x$	27. $\dfrac{d}{dx}(\tanh x)=\operatorname{sech}^{2}x$	36. $\dfrac{d}{dx}(\coth^{-1}x)=\frac{1}{1-x^{2}}$

Integration Rules

11. $\displaystyle\int c\cdot f(x)\ dx=c\int f(x)\ dx$	11. $\displaystyle\int\tan x\ dx=-\ln\left\lvert\cos x\right\rvert+C$	23. $\displaystyle\int\frac{1}{x\sqrt{x^{2}-a^{2}}}\ dx=\frac{1}{a}\sec^{-1}\left(% \frac{\left\lvert x\right\rvert}{a}\right)+C$
12. $\displaystyle\int f(x)\pm g(x)\ dx=$	12. $\displaystyle\int\sec x\ dx=\ln\left\lvert\sec x+\tan x\right\rvert+C$	24. $\displaystyle\int\cosh x\ dx=\sinh x+C$
12. $\displaystyle\int f(x)\ dx\pm\int g(x)\ dx$	13. $\displaystyle\int\csc x\ =-\ln\left\lvert\csc x+\cot x\right\rvert+C$	25. $\displaystyle\int\sinh x\ dx=\cosh x+C$
13. $\displaystyle\int 0\ dx=C$	14. $\displaystyle\int\cot x\ dx=\ln\left\lvert\sin x\right\rvert+C$	26. $\displaystyle\int\tanh x\ dx=\ln(\cosh x)+C$
14. $\displaystyle\int 1\ dx=x+C$	15. $\displaystyle\int\sec^{2}x\ dx=\tan x+C$	27. $\displaystyle\int\coth x\ dx=\ln\left\lvert\sinh x\right\rvert+C$
1 5. $\displaystyle\int{}x^{n}\ dx=\frac{1}{n+1}x^{n+1}+C$ , $n\neq-1$	16. $\displaystyle\int\csc^{2}x\ dx=-\cot x+C$	28. $\displaystyle\int\frac{1}{\sqrt{x^{2}-a^{2}}}\ dx=\ln\left\lvert x+\sqrt{x^{2}% -a^{2}}\right\rvert+C$
16. $\displaystyle\int e^{x}\ dx=e^{x}+C$	17. $\displaystyle\int\sec x\tan x\ dx=\sec x+C$	29. $\displaystyle\int\frac{1}{\sqrt{x^{2}+a^{2}}}\ dx=\ln\left\lvert x+\sqrt{x^{2}% +a^{2}}\right\rvert+C$
17. $\displaystyle\int a^{x}\ dx=\frac{1}{\ln a}\cdot a^{x}+C$	18. $\displaystyle\int\csc x\cot x\ dx=-\csc x+C$	30. $\displaystyle\int\frac{1}{a^{2}-x^{2}}\ dx=\frac{1}{2a}\ln\left\lvert\frac{a+x% }{a-x}\right\rvert+C$
18. $\displaystyle\int\frac{1}{x}\ dx=\ln\left\lvert x\right\rvert+C$	19. $\displaystyle\int\cos^{2}x\ dx=\frac{1}{2}x+\frac{1}{4}\sin\bigl{(}2x\bigr{)}+C$	31. $\displaystyle\int\frac{1}{x\sqrt{a^{2}-x^{2}}}\ dx=\frac{1}{a}\ln\left(\frac{x% }{a+\sqrt{a^{2}-x^{2}}}\right)+C$
19. $\displaystyle\int\cos x\ dx=\sin x+C$	20. $\displaystyle\int\sin^{2}x\ dx=\frac{1}{2}x-\frac{1}{4}\sin\bigl{(}2x\bigr{)}+C$	32. $\displaystyle\int\frac{1}{x\sqrt{x^{2}+a^{2}}}\ dx=\frac{1}{a}\ln\left\lvert% \frac{x}{a+\sqrt{x^{2}+a^{2}}}\right\rvert+C$
10. $\displaystyle\int\sin x\ dx=-\cos x+C$	21. $\displaystyle\int\frac{1}{x^{2}+a^{2}}\ dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{% a}\right)+C$	33. $\displaystyle\int\sqrt{x^{2}+a^{2}}\ dx=$
	22. $\displaystyle\int\frac{1}{\sqrt{a^{2}-x^{2}}}\ dx=\sin^{-1}\left(\frac{x}{% \left\lvert a\right\rvert}\right)+C$	33. $\displaystyle\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}\ln\left(x+\sqrt{x^{% 2}+a^{2}}\right)+C$

The Unit Circle

Definitions of the Trigonometric Functions

Unit Circle Definition

\displaystyle\sin\theta

\displaystyle=y

\displaystyle\cos\theta

\displaystyle=x

\displaystyle\csc\theta

\displaystyle=\dfrac{1}{y}

\displaystyle\sec\theta

\displaystyle=\dfrac{1}{x}

\displaystyle\tan\theta

\displaystyle=\frac{y}{x}

\displaystyle\cot\theta

\displaystyle=\frac{x}{y}

Right Triangle Definition

\displaystyle\sin\theta

\displaystyle=\frac{\text{O}}{\text{H}}

\displaystyle\csc\theta

\displaystyle=\frac{\text{H}}{\text{O}}

\displaystyle\cos\theta

\displaystyle=\frac{\text{A}}{\text{H}}

\displaystyle\sec\theta

\displaystyle=\frac{\text{H}}{\text{A}}

\displaystyle\tan\theta

\displaystyle=\frac{\text{O}}{\text{A}}

\displaystyle\cot\theta

\displaystyle=\frac{\text{A}}{\text{O}}

Common Trigonometric Identities

Pythagorean Identities

	$\displaystyle\sin^{2}x+\cos^{2}x=1$
	$\displaystyle\tan^{2}x+1=\sec^{2}x$
	$\displaystyle 1+\cot^{2}x=\csc^{2}x$

Cofunction Identities

$\displaystyle\sin\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\cos x$	$\displaystyle\csc\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\sec x$
$\displaystyle\cos\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\sin x$	$\displaystyle\sec\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\csc x$
$\displaystyle\tan\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\cot x$	$\displaystyle\cot\left(\frac{\pi}{2}-x\right)$	$\displaystyle=\tan x$

Double Angle Formulas

	$\displaystyle\sin 2x$	$\displaystyle=2\sin x\cos x$
	$\displaystyle\cos 2x$	$\displaystyle=\cos^{2}x-\sin^{2}x$
		$\displaystyle=2\cos^{2}x-1$
		$\displaystyle=1-2\sin^{2}x$
	$\displaystyle\tan 2x$	$\displaystyle=\frac{2\tan x}{1-\tan^{2}x}$

Sum to Product Formulas

	$\displaystyle\sin x+\sin y$	$\displaystyle=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
	$\displaystyle\sin x-\sin y$	$\displaystyle=2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)$
	$\displaystyle\cos x+\cos y$	$\displaystyle=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
	$\displaystyle\cos x-\cos y$	$\displaystyle=2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{y-x}{2}\right)$

Power–Reducing Formulas

	$\displaystyle\sin^{2}x$	$\displaystyle=\frac{1-\cos 2x}{2}$
	$\displaystyle\cos^{2}x$	$\displaystyle=\frac{1+\cos 2x}{2}$
	$\displaystyle\tan^{2}x$	$\displaystyle=\frac{1-\cos 2x}{1+\cos 2x}$

Even/Odd Identities

	$\displaystyle\sin(-x)$	$\displaystyle=-\sin x$
	$\displaystyle\cos(-x)$	$\displaystyle=\phantom{-}\cos x$
	$\displaystyle\tan(-x)$	$\displaystyle=-\tan x$
	$\displaystyle\csc(-x)$	$\displaystyle=-\csc x$
	$\displaystyle\sec(-x)$	$\displaystyle=\phantom{-}\sec x$
	$\displaystyle\cot(-x)$	$\displaystyle=-\cot x$

Product to Sum Formulas

	$\displaystyle\sin x\sin y$	$\displaystyle=\frac{1}{2}\big{(}\cos(x-y)-\cos(x+y)\big{)}$
	$\displaystyle\cos x\cos y$	$\displaystyle=\frac{1}{2}\big{(}\cos(x-y)+\cos(x+y)\big{)}$
	$\displaystyle\sin x\cos y$	$\displaystyle=\frac{1}{2}\big{(}\sin(x+y)+\sin(x-y)\big{)}$

Angle Sum/Difference Formulas

	$\displaystyle\sin(x\pm y)$	$\displaystyle=\sin x\cos y\pm\cos x\sin y$
	$\displaystyle\cos(x\pm y)$	$\displaystyle=\cos x\cos y\mp\sin x\sin y$
	$\displaystyle\tan(x\pm y)$	$\displaystyle=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y}$

Areas and Volumes

Triangles $\displaystyle h=a\sin\theta$ $\displaystyle\text{Area}=\frac{1}{2}bh$ Law of Cosines: $\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos\theta$		Right Circular Cone $\displaystyle\text{Volume}=\frac{1}{3}\pi r^{2}h$ $\displaystyle\text{Surface Area}=$ $\displaystyle\pi r\sqrt{r^{2}+h^{2}}+\pi r^{2}$
Parallelograms Area = $b h$		Right Circular Cylinder $\displaystyle\text{Volume}=\pi r^{2}h$ $\displaystyle\text{Surface Area}=$ $\displaystyle 2\pi rh+2\pi r^{2}$
Trapezoids Area = $\frac{1}{2}(a+b)h$		Sphere $\displaystyle\text{Volume}=\frac{4}{3}\pi r^{3}$ $\displaystyle\text{Surface Area}=4\pi r^{2}$
Circles $\displaystyle\text{Area}=\pi r^{2}$ $\displaystyle\text{Circumference}=2\pi r$		General Cone $\displaystyle\text{Area of Base}=A$ $\displaystyle\text{Volume}=\frac{1}{3}Ah$
Sectors of Circles $\displaystyle\theta\text{ in radians}$ $\displaystyle\text{Area}=\frac{1}{2}\theta r^{2}$ $\displaystyle s=r\theta$		General Right Cylinder $\displaystyle\text{Area of Base}=A$ $\displaystyle\text{Volume}=Ah$

Algebra

Factors and Zeros of Polynomials

Let $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0}$ be a polynomial. If $p(a)=0$ , then $a$ is a $z e r o$ of the polynomial and a solution of the equation $p(x)=0$ . Furthermore, $(x-a)$ is a $f a c t o r$ of the polynomial.

Fundamental Theorem of Algebra

An $n$ th degree polynomial has $n$ (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.

Quadratic Formula

If $p(x)=ax^{2}+bx+c$ , then the zeros of $p$ are $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Special Factoring

\displaystyle x^{2}-a^{2}

\displaystyle=(x-a)(x+a)

\displaystyle x^{3}\pm a^{3}

\displaystyle=(x\pm a)(x^{2}\mp ax+a^{2})

\displaystyle x^{4}-a^{4}

\displaystyle=(x^{2}-a^{2})(x^{2}+a^{2})

Binomial Theorem

	$\displaystyle(x+y)^{2}$	$\displaystyle=x^{2}+2xy+y^{2}$	$\displaystyle(x+y)^{3}$	$\displaystyle=x^{3}+3x^{2}y+3xy^{2}+y^{3}$
	$\displaystyle(x+y)^{4}$	$\displaystyle=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}$	$\displaystyle(x+y)^{n}$	$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}x\hskip 0.75pt^{n-k}y\hskip 0.75pt^{k}$

Rational Zero Theorem

If $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb+a_{1}x+a_{0}$ has integer coefficients, then every $r a t i o n a l$ $z e r o$ of $p$ is of the form $x=r/s$ , where $r$ is a factor of $a_{0}$ and $s$ is a factor of $a_{n}$ .

Factoring by Grouping

$acx^{3}+adx^{2}+bcx+bd=ax^{2}(cx+d)+b(cx+d)=(ax^{2}+b)(cx+d)$

Arithmetic Operations

$\displaystyle ab+ac=a(b+c)$	$\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$	$\displaystyle\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
$\displaystyle\frac{\left(\dfrac{a\vphantom{d}}{b}\right)}{\left(\dfrac{c}{d}% \right)}=\left(\frac{a}{b}\right)\left(\frac{d}{c}\right)=\frac{ad}{bc}$	$\displaystyle\frac{\left(\dfrac{a}{b}\right)}{c}=\frac{a}{bc}$	$\displaystyle\frac{a}{\left(\dfrac{b}{c}\right)}=\frac{ac}{b}$
$\displaystyle a\left(\frac{b}{c}\right)=\frac{ab}{c}$	$\displaystyle\frac{a-b}{c-d}=\frac{b-a}{d-c}$	$\displaystyle\frac{ab+ac}{a}=b+c$

Exponents and Radicals

\displaystyle a^{0}=1,\;\;a\neq 0

\displaystyle(ab)^{x}

\displaystyle=a^{x}b^{x}

\displaystyle a^{x}a^{y}

\displaystyle=a^{x+y}

\displaystyle\sqrt{a}

\displaystyle=a^{1/2}

\displaystyle\frac{a^{x}}{a^{y}}

\displaystyle=a^{x-y}

\displaystyle\sqrt[n]{a}

\displaystyle=a^{1/n}

\displaystyle\left(\frac{a}{b}\right)^{x}=\frac{a^{x}}{b^{x}}

\displaystyle\sqrt[n]{a^{m}}

\displaystyle=a^{m/n}

\displaystyle a^{-x}

\displaystyle=\frac{1}{a^{x}}

\displaystyle\sqrt[n]{ab}

\displaystyle=\sqrt[n]{a}\sqrt[n]{b}

\displaystyle(a^{x})^{y}

\displaystyle=a^{xy}

\displaystyle\sqrt[n]{\frac{a}{b}}

\displaystyle=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Additional Formulas

Summation Formulas

\displaystyle\sum^{n}_{i=1}{c}

\displaystyle=cn

\displaystyle\sum^{n}_{i=1}{i}

\displaystyle=\frac{n(n+1)}{2}

\displaystyle\sum^{n}_{i=1}{i^{2}}

\displaystyle=\frac{n(n+1)(2n+1)}{6}

\displaystyle\sum^{n}_{i=1}{i^{3}}

\displaystyle=\left(\frac{n(n+1)}{2}\right)^{2}

Trapezoidal Rule

$\displaystyle\int_{a}^{b}{f(x)}\ dx\approx\frac{\Delta x}{2}\left[f(x_{1})+2f(% x_{2})+2f(x_{3})+\dotsb+2f(x_{n})+f(x_{n+1})\right]$
with $\text{Error}\leq\dfrac{(b-a)^{3}}{12n^{2}}\left[\max\left\lvert f\,^{\prime% \prime}(x)\right\rvert\right]$

Simpson’s Rule

$\displaystyle\int_{a}^{b}{f(x)}\ dx\approx\frac{\Delta x}{3}\left[f(x_{1})+4f(% x_{2})+2f(x_{3})+4f(x_{4})+\dotsb+2f(x_{n-1})+4f(x_{n})+f(x_{n+1})\right]$
with $\text{Error}\leq\dfrac{(b-a)^{5}}{180n^{4}}\left[\max\left\lvert f\hskip 0.75% pt^{(4)}(x)\right\rvert\right]$

Arc Length

$\displaystyle L=\int_{a}^{b}{\sqrt{1+f\,^{\prime}(x)^{2}}}\ dx$

Work Done by a Variable Force

$\displaystyle W=\int_{a}^{b}{F(x)}\ dx$

Force Exerted by a Fluid

$\displaystyle F=\int_{a}^{b}{w\,d(y)\,\ell(y)}\ dy$

Taylor Series Expansion for $f(x)$

$\displaystyle p_{n}(x)=f(c)+f\,^{\prime}(c)(x-c)+\frac{f\,^{\prime\prime}(c)}{% 2!}(x-c)^{2}+\frac{f\,^{\prime\prime\prime}(c)}{3!}(x-c)^{3}+\dotsb+\frac{f\,^% {(n)}(c)}{n!}(x-c)^{n}+\dotsb$

Standard Form of Conic Sections

Parabola		Ellipse	Hyperbola
Vertical axis	Horizontal axis		Foci and vertices	Foci and vertices
			on $x$ -axis	on $y$ -axis
$y=\dfrac{x^{2}}{4p}$	$x=\dfrac{y^{2}}{4p}$	$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$	$\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$	$\dfrac{y^{2}}{b^{2}}-\dfrac{x^{2}}{a^{2}}=1$

Summary of Tests for Series

Test	Series	Condition(s) of Convergence	Condition(s) of Divergence	Comment
$n^{\text{th}}$ -Term Test for Divergence	$\displaystyle\sum_{n=1}^{\infty}a_{n}$		$\displaystyle{\lim_{n\to\infty}a_{n}\neq 0}$	cannot show convergence.
Alternating Series	$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}b_{n}$	$\displaystyle\lim_{n\to\infty}b_{n}=0$		$b_{n}$ must be positive and decreasing
Geometric Series	$\displaystyle\sum_{n=0}^{\infty}ar\hskip 0.75pt^{n}$	$\left\lvert r\right\rvert<1$	$\left\lvert r\right\rvert\geq 1$	Sum $=\dfrac{a}{1-r}$
Telescoping Series	$\displaystyle\sum_{n=1}^{\infty}b_{n}-b_{n+m}$	$\displaystyle{\lim_{n\to\infty}b_{n}=L}$		Sum $=$ $\displaystyle\left(\sum_{n=1}^{m}b_{n}\right)-L$
$p$ -Series	$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)^{p}}$	$p>1$	$p\leq 1$
$p$ -Series For Logarithms	$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)(\log n)^{p}}$	$p>1$	$p\leq 1$	logarithm’s base doesn’t affect convergence.
Integral Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$\displaystyle\int_{1}^{\infty}a(n)\ dn$ convergesd	$\displaystyle\int_{1}^{\infty}a(n)\ dn$ diverges	$a_{n}=a(n)$ must be positive and decreasing
Direct Comparison	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$\displaystyle\sum_{n=0}^{\infty}b_{n}$ converges and $0\leq a_{n}\leq b_{n}$	$\displaystyle\sum_{n=0}^{\infty}b_{n}$ diverges and $0\leq b_{n}\leq a_{n}$
Limit Comparison	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$\displaystyle\sum_{n=0}^{\infty}b_{n}$ converges and $\displaystyle\lim_{n\to\infty}a_{n}/b_{n}<\infty$	$\displaystyle\sum_{n=0}^{\infty}b_{n}$ diverges and $\begin{aligned} \displaystyle\lim_{n\to\infty}a_{n}/b_{n}&\displaystyle>0\\ \displaystyle\text{or }&\displaystyle=\infty\end{aligned}$	$a_{n},b_{n}>0$
Ratio Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$\displaystyle\lim_{n\to\infty}\left\lvert\frac{a_{n+1}}{a_{n}}\right\rvert<1$	$\begin{aligned} \displaystyle\lim_{n\to\infty}\left\lvert\frac{a_{n+1}}{a_{n}}% \right\rvert&\displaystyle>1\\ \displaystyle\text{or }&\displaystyle=\infty\end{aligned}$	limit of 1 is indeterminate
Root Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$\displaystyle\lim_{n\to\infty}\left\lvert a_{n}\right\rvert^{1/n}<1$	$\begin{aligned} \displaystyle\lim_{n\to\infty}\left\lvert a_{n}\right\rvert^{1% /n}&\displaystyle>1\\ \displaystyle\text{or }&\displaystyle=\infty\end{aligned}$	limit of 1 is indeterminate