CALCULUS 2 - LECTURE, CH 12
12.1 Sequences Hwk: Std (1, 6, 9, 14, 17, …) to 61, +5, -46
Def: a function whose domain is the set of natural numbers (or whole numbers)
Ex built on f(x)=x2: we might write f(n) = n2, n = 1, 2, 3, ...
D: {1, 2, 3, 4, 5, …} term numbers
R: {1, 4, 9, 16, 25, …} terms
Compare the graph of the sequence with the graph of the continuous function
Vocab and notation:
The sequence is usually thought of as the range values (domain is not referred to explicitly)
We use n for an element of the domain, then an is the corresp element of the sequence a.
Ex: a3 = 9 because n = 3,
{an} or 
refer to the entire
sequence [w provisions to stress or modify the starting point]
Ex: std notation for our sequence would be {an} = {n2}
Task: find first several terms of a series:
Ex: 
= …{2/1, 3/2, 4/3, …}
Ex: 
= {1, 0, -1,
0, 1, 0, -1, 0, …}
Does a sequence have a limit (converge) or not (diverge)?
Ex: limit of x/(x-1) is 1 as x appr ¥,
so 
converges to 1
Def: [explain graphically] For every epsilon>0, there exists N>0 such that whenever n>N,
abs(an-L)<epsilon
Relate to the example series above.
See properties of limits
Squeeze Theorem applies
Ex: -1/n £ cos(pn)/n £ 1/n
Thm: if |an|→0, then an→0
Convenient thm: if a seq is monotonic and bounded, then it converges
Defs
How can you show that a sequence is decreasing (increasing)/
Show an+1 < an, or
Show an / an+1 <1, or
Show deriv is always negative
Illustration for 
--Since it's
monotonic decreasing, 1st term is a upper bound, and since both num and denon
are pos, 0 is a convenient choice for a lower bound (after you find the
limit, you can conclude that 1 is the greatest lower bound, but any lower bound
is enough for the thm.)
12.2 Series Hwk: Std to 66, +3, 11, 42, -26, 34
Intro: Induction proofs
Prove for n=1 (or initial value);
Assume true for n=k [assuming that there is a highest index for which the proposition is valid]; Prove true for n=k+1 [so there is no such highest valid level]
Examples (relevant for series work):
, 
Applic: 1/2 +1/4 + 1/8 + …+1/64 = 1/2 * (1 - (1/2)^6) / (1-1/2) = 63/64
Take limits: Geom series converges to a / (1-r) if |r|<1.
Inf sum of 1/2 +1/4 + 1/8 + … is 1
Violate condition: 1+2+4+8+… = -1
Series = sum of terms of a sequence, actually a series is a sequence of partial sums, where a partial sum is a "running total," sum of terms of {an} from 1 to n, as n increases from 1 to infinity.
12.3 Integral Test and Estimates of Sums Hwk: 3, 5, 8, 13, 15, 18, 21, 25, 28, 30
S = Sn + Rn
For decreasing-term positive-term series, int(f, x,n+1,¥) £ Rn £ int(f,x,n,¥)
The first rectangle in Rn (i.e.an+1)
is pictured twicein the figure below:
the integral of the continuous
function beginning at n is above the rectangle,
and the integral beginning at
n+1 is below the rectangle
12.4 Comparison Tests Hwk: Std to 29, +19, 31, 33
Basic Comparison Test:
Let an £ bn for every n³N
If the bigger series converges, then the smaller series converges
If the smaller series diverges, then the bigger series diverges
Ex: Test Sum of 1 / (2n+n), Sum of 1 / (sqrt(n) - 1)
Problem: The result is the same if the signs in the denominator are reversed, but this test doesn't show it.
Limit Comparison Test:
Calculate the limit of the ratio of the nth terms for two series, one with a known convergence result.
If the limit is finite and not 0, both series have the same convergence result.
The comparison test is often a p-series or a geometric series.
Ex: The same series above, with the opposite signs in the denominator.
12.5 Alternating Series Hwk: 1, 2, 3, 4, 6, 9, 14, 17, 23, 26, 29
Requirements for convergence:
successive terms decrease in abs value
nth term, in abs value, has limit of 0
Ex: alt harm series
Est of sums: max error: abs(Rn) = abs(s-sn) £ abs(an+1)
How many terms are required to get a given level of accuracy?
12.6 Absolute Convergence; Ratio and Root Tests Hwk: 1, 6, 7, 9, 15, 16, 21, 22, 25, 29, 34
12.7 OMIT
12.8 Power Series Hwk: Std to #26, +2, 7, 16, 24
Finding the radius and interval of convergence
12.9 Representations of Functions as Power Series Hwk: Std [to 37] +23, 30, -17
Extra [func in #23] Use of the series for ln((1-x) / (1+x)) to find ln x when x is greater than 2.
Extra: Bessel funcs solve DEs -- See #35, 36, see intro page 726, see discussion page 775 on apps (planetary motion, temp distrib in a circular plate, shape of vibrating drumhead)
Demo: method of Frobenius on J0, problem 35
12.11 Binomial Series Hwk:
Identify families of functions
1. ex to sin, cos, sinh, cosh
identity reiq = rcisq
2. binomial to geometric to
a. log
b. arctan
c. arcsin
Return to: Merced College; Don Power Updated 01/16/03 by Don Power