Chapter 9 notes

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9.3 Sequences - Defining a sequence explicitly(as a function of the term number, usually n or k.

an=n^2-5n_this is the rule.

a1=-4

a2=-6

a3=-6

a4=-4

a5=0

a6=6

9.3 Sequences - Defining a sequence recursively(as a function of the previous terms)

Find the first 4 terms and the eighth term for b1=-2, b2=3 and bk=bk-2+bk-1

b1=-2 b2=3 b3=1 b4=4 b5=5 b6=9 b7=14 b8=23

9.3 Sequences - Finding limits of sequences; Convergence and divergence

Find the limit of the sequences; Determine whether it converges or diverges.

a) 1/1, 1/2, 1/3, 1/4...,1/n

lim 1/n=0 converges to 0

b) 2/1, 3/2, 4/3, 5/4 ...

lim n+1/n=1 converges to 1

c) -1,1,-1,1 ... (-1)^n

Lim(-1)^n=1 or -1 "oscillating" going back and forth, falls under divergence.

Arithmetic sequences; have a common difference between each pair of consecutive terms. The nth term is given as

an=a1+(n-1)d or an=an-1+d

(explicitly) (recursively)

Geometric sequences; Have a common ratio between each pair of consecutive terms. The nth term is given as

an=a1r^n-1 or an=an-1r

(explicitly) (recursively)