calc ii notes (sequences/series)

geometric sequence formula

a_n = a₁r^n-1

infinite sum formula

S_n= (a₁) 1/1-r

when given two a values what formula should be used?

a_n/a_m = r^n-m

when given two partial sums what formula can be used to find a_n?

a_n= S_n - S_n-1

telescoping

two fractions that increase by r each time.

finite sum formula

S_n = (a1)(1-rⁿ)/1-r

divergence/convergence of a sequence: How do you know whether a sequence diverges?

find the limit. If the limit is positive/negative infinity diverges. if it bounces between 2 numbers, it diverges. If the limit does not exist.

divergence/convergence of a series: How do you know whether a series diverges?

find the limit of the series if it doesn’t equal 0 it diverges. if it equal to 0 use other tests to find a conclusion