Computer Science · Data Sheet

Finite Calculus and Power Series Cheat Sheet

Symbol	Constant	Value	Unit
Delta f(x)	Forward difference	Delta f(x)=f(x+1)-f(x)	formula
E f(x)	Shift operator	E f(x)=f(x+1)	formula
Delta(uv)	Difference product rule	Delta(uv)=u Delta(v)+E(v) Delta(u)	formula
Delta x_falling_n	Difference power rule	Delta x_falling_n=n*x_falling_(n-1)	formula
1/(1-x)	Geometric power series	1/(1-x)=sum(i=0..infinity)x^i	formula
exp(x)	Exponential series	exp(x)=sum(i=0..infinity)x^i/i!	formula
ln(1+x)	Logarithm series	ln(1+x)=sum(i=1..infinity)(-1)^(i+1)x^i/i	formula
sin(x)	Sine series	sin(x)=sum(i=0..infinity)(-1)^i*x^(2i+1)/(2i+1)!	formula
cos(x)	Cosine series	cos(x)=sum(i=0..infinity)(-1)^i*x^(2i)/(2i)!	formula
arctan(x)	Arctangent series	arctan(x)=sum(i=0..infinity)(-1)^i*x^(2i+1)/(2i+1)	formula
(x+y)^n	Binomial theorem	(x+y)^n=sum(k=0..n)C(n,k)x^(n-k)y^k	formula
x^n-y^n	Difference of powers	x^n-y^n=(x-y)sum(k=0..n-1)x^(n-1-k)y^k	formula
A(x)	Ordinary generating function	A(x)=sum(i=0..infinity)a_i*x^i	formula
A(x)	Exponential generating function	A(x)=sum(i=0..infinity)a_i*x^i/i!	formula
A'(x)	Generating function derivative	A'(x)=sum(i=0..infinity)(i+1)a_(i+1)x^i	formula
A(x)B(x)	Generating function convolution	[x^i]A(x)B(x)=sum(j=0..i)a_j*b_(i-j)	formula

About this calculator

A searchable formula reference transcribed and normalized from the supplied Theoretical Computer Science Cheat Sheet.

Finite Calculus and Power Series formula reference

This searchable reference organizes formulas from the supplied Theoretical Computer Science Cheat Sheet into web-friendly notation. Use the search box to find a formula by name or symbol, then review its assumptions before applying it.

Some identities use conventional mathematical notation such as summation limits, factorials, congruences, expectations and asymptotic bounds. These entries are reference material and should be interpreted within their stated domains.