Calc Week 2

Question 1

Absolute Value Theorem

Answer 1

Let {a sub n} be a sequence.
if lim n -> inf |a sub n|= 0;
then lim n-> inf a sub n= 0
—
if {|a sub n|} is a null sequence
then {a sub n} is a null sequence too

Question 2

Monotone convergence theorem

Answer 2

if {a sub n} is both monotonically increasing and bounded above, then {a sub n} converges

if {a sub n} is both monotonically decreasing and bounded below, the {a sub n} converges

• does not tell you to what, but that it does

Question 3

the connection

Answer 3

the sum of the sequence (infinite) =
lim n-> inf Sn
—
the sum of the sequence (infinite) should equal the limit of the partial sum
—-
q: does the sequence converge?
new q: does the sequence of partial sums converge?

Question 4

telescoping sum

Answer 4

2 - 1/n

• terms cancel

Question 5

infinite series

Answer 5

let {a sub n} be a sequence and let {s sub n} be its sequence of partial sums then:

1. the infinite sum of sigma from n=1 to inf is called an infinite series, or series
2. if {s sub n} converges to L, then we say that the series sigma from n=1 to inf of a sub n converges to L so we write sigma from n=1 to inf of a sub n = L
3. if {s sub n} diverges we say that (the series) sigma from n=1 to inf of a sub n diverges

Question 6

remark about the connection

Answer 6

if we let s sub n = sigma from k = 1 to inf of a sub k = a1 +…+a sub n
then we have:

sigma from k = 1 to inf of a sub k =
lim n -> inf (sigma from k=1 to n of a sub k) =
lim n-> inf s sub n

so questions about the convergence of sigma k=1 to inf of a sub k are really questions about the convergence of {s sub n}

Question 7

oscillating series

Answer 7

sigma from n=1 to inf of (-1) ^n

Question 8

does a series converge

Answer 8

if {s sub n} converges, then so does the series of a sub n
when {s sub n} is {a sub n}’s partial sum

Question 9

remark for a series to converge, s sub n must

Answer 9

for a series sigma n=1 to inf of a sub n to converge say it equals L (converges to L), we must have s sub n converge to L

Question 10

n-th term test

Answer 10

consider the series sigma from n=1 to inf of a sub n.

if lim n-> inf of a sub n != 0 then
(the series) sigma from n = 1 to inf of a sub n diverges

if , the series, sigma from n =1 to inf of a sub n converges, then lim n-> inf of a sub n = 0

Question 11

remark of warning for n-th term test

Answer 11

there are series for which a sub n -> 0 but the series diverges

Question 12

power series expansion of e^x

Answer 12

for any number x, the series
sigma from n=0 to inf of x^n/!n is convergent

converges to e^x
so we can write

sigma from n=0 to inf od x^n/!n = e^x

because it converges, it follow by the n-th term test that
lim n->inf of x^n/!n = 0
- this limit can be summarized as saying factorial beats exponential

Question 13

a geometric series

Answer 13

a series of the form
sigma from n=0 to inf of r^n = 1 + r+ r^2 + r^3+….

Question 14

remarking of indexing

Answer 14

another way of writing the same series

example of writing a geometric series differently

sigma from n=1 to inf of r ^n-1 = 1 + r+ r^2+…

Question 15

look over the problems in these lectures

Answer 15

:)