Sequences
a sequence is a function which takes the natural numbers as its domain
Monotonic Sequence
A sequence that is increasing or decreasing
Geometric Series Formula
Sn=a1(1-r^n)/(1-r)
Telescopic Series
see pic
Divergence Test
lim_(k->infinity) [a_k] doesn't equal zero, then the series Σa_k must diverge
Harmonic Series
diverges
Comparison Test
If 0≤An≤bn and if ∑bn converges, then ∑An converges
Limit Comparison Test
if lim as n approaches ∞ of ratio of comparison series/general term is positive and finite, then series behaves like comparison series
Alternating Series
∑(from n=1 to infinity) (-1)^n-1 a{n}. Converges if 0<a{n+1}<a{n} and lim (as n approaches infinity) a{n}= 0
Alternating Series Test
lim as n approaches zero of general term = 0 and terms decrease, series converges
Absolute Convergence Test
If the sum of |a[n]| converges, then the sum of a[n] converges.
Ratio Test
lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges
Power Series
A series which represents a function as a polynomial that goes on forever and has no highest power of x.
Root Test
limit as n approaches infinity of the nth root of the series, if <1 converges absolutely, >1 diverges, =1 no conclusion can be drawn
The Taylor Series
f(c)+f'(c)(x-c)+f''(c)(x-c)^2/2 ... +f^n(c)(x-c)^n/n!
Maclaurin Series
(f^(n)(0)/n!) (x)^n
Binomial Series
1+Σ[∞ to n=1] (k(k-1)(k-2)(k-3)...(k-n+1))/n! (x^n)