Calculus Early Transcendentals / Ch.11


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


Telescopic Series

see pic

Divergence Test

lim_(k->infinity) [a_k] doesn't equal zero, then the series Σa_k must diverge

Harmonic Series


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)