Calc Week 5 Midterm Study

Question 1

Evaluating the limit of a sequence

Answer 1

1. by calculation and algebraic limit theorem
   -multiply numerator and denominator by 1/ biggest term of all
   -simplify
   -take the limit
2. thinking of f(x)
   -set a sub n to an equation
   -set f(x) = a sub n with x plugged in
   -take limit of f(x) usually involves L’H
   -answer will be lim n-> inf of a sub n
3. multiplying conjugates
   -an = equation of the sequence
   -multiply by a fraction (numerator and denominator are that same a sub n but opposite sign)
   -simplify
   -take limit n -> inf

Question 2

writing the partial sums of a series

Answer 2

the next s sub n includes the previous s sub n plus the current s sub n

Question 3

the connection between partial sums and infinite series is

Answer 3

lim n-> inf s sub n =
the sum of the series a sub n

Question 4

geometric series test
-when to use
-definition
-how to use
-explanation

Answer 4

for a geometric series
r^n

1.the nth partial sum is s sub n = 1-r^n+1/1-r
2. the series converges if and only if |r| < 1 to 1/1-r

1. make sure it is a geometric series
   2.look at r
   3.use test

since |r| < 1, the series converges to 1/1-r by the geometric series test
since |r|>=1, the series diverges by the geometric series test

Question 5

p-series test
-when to use
-definition
-how to use
-explanation

Answer 5

for a p-series
1/n^p for 1/(an+b)^p, p>0

a general p-series converges if and only p>1

1.make sure it is a p-series
2. look at p
3. use test

since p>1, the series converges by the p-series test
since p<=1, the series diverges by the p-series test

Question 6

direct comparison test
-when to use
-definition
-how to use
-explanation

Answer 6

for when you can find another series that is larger or smaller that you know converges or diverges

suppose {a sub n} and {b sub n} satisfy o<=a sub n<= b sub n for all n then
1.if series b sub n converges, then series a sub n converges
2.if series a sub n diverges, then series b sub n diverges

1. find another series you know that converges or diverges
2. figure out what the new series does and if it larger or smaller than original
3. use test

since series a sub n or b sub n converges or diverges, the direct comparison test implies series a sub n or b sub n converge or diverge

Question 7

limit comparison test
-when to use
-definition
-how to use
-explanation

Answer 7

for when you can find a similar/comparable series that you know converges or diverges

let {a sub n} and {b sub n} be sequences such that a sub n>= 0 and b sub n>0 for all n then
1.if lim n-> inf a sub n/b sub n =L, with 0<L<inf the a sub n series converges if and only if the series b sub n converges (works for divergence too)
2.if lim n-> inf a sub n/b sub n = 0 and series b sub n < inf then series a sub n < inf
3. if lim n-> inf a sub n/b sub n= +inf and series b sub n = +inf then series a sub n = +inf

1.find b sub n similar to a sub n
2.take the lim of a sub n /b sub n to see if comparable
3.use test

since L is finite and positive, series a sub n and b sub n are comparable, therefore, the limit comparison test implies since series b sub n converges, so does a sub n

Question 8

ratio test
-when to use
-definition
-how to use
-explanation

Answer 8

when you see factorials

let {a sub n} be a positive sequence with lim n-> inf a sub n+1/a sub n = L then
1.if L<1, then series a sub n converges
2.if L>1, then series a sub n diverges
3.if L=1, the test is inconclusive

1. find a sub n
   2.take lim n-> of a sub n+1/a sub n
   —plug in n+1 and n into a sub n
2. use test

since L = # >/< 1, by the ration test, the series converges/diverges

Question 9

alternating series test
-when to use
-definition
-how to use
-explanation

Answer 9

for an alternating series
(-1)^n (a sub n)
(-1)^n+1(a sub n)
–a sub n is positive

let {a sub n} be a positive, monotonically decreasing sequence such that lim n-> inf a sub n = 0 then the series converges

1.find a sub n
2.is a sub n monotonically decreasing
3.is the lim n-> inf of a sub n =0
4.use test

since a sub n is monotonically decreasing and lim n-> inf = 0, then the series converges by the alternating series test

Question 10

n-term test
-when to use
-definition
-how to use
-explanation

Answer 10

first thing to check to see if automatically diverge

consider the series of a sub n
if lim n-> inf a sub n != 0, the series of a sub n diverges

1. find a sub n
2. take the lim n-> inf a sub n

since the lim n-> inf of a sub n does not equal 0, the series converges by the n-th term test

Question 11

other things to know
lim n-> of
(1+a/n)^n

Answer 11

e^a

Question 12

other things to know
lim x->0 of sinx/x

Answer 12

1

Question 13

other things to know
absolute value theorem

Answer 13

for a sequence

let {a sub n} be a sequence
if lim->inf of |a sub n |=0 then lim n->inf of a sub n =0

Question 14

other things to know
power series expansion of e^x

Answer 14

series from n=1 to inf of
x^n/n!
=
e^x

factorial beats exponential

Question 15

other things to know
absolute convergence theorem

Answer 15

suppose series of |a sub n| converges, then
1.series a sub n converges
2.the value of series of a sub n does not depend on the order of the terms

Question 16

other things to know
definition of converging absolutely or conditionally

Answer 16

must know if the series even converges
must converge for either one to be true

let series of a sub n be any series
1. if series |a sub n| converges, the series converges absolutely
2.if series a sub n converges, but series |a sub n| = +inf then the series converges conditionally

Question 17

review

Answer 17

7 tests and when to use, definition, steps and explanation

evaluating a sequence

writing partial sums

the connection

the limit/or sum of 3 (fact) series

3 theorems

——-other things to know—–

algebraic limit theorem

convergence theorem

monotone convergence theorem

integral test

showing comparability (in limit comparison test)

root test