Math 115 Calculus II

Math 115 Calculus II

11.1 Sequences, 11.2 Series, 11.3 The Integral Test, 11.4 Comparison Tests

Mikey Chow

March 16, 2021

Table of Contents

Notation for Sequences

Using Patterns to Write Formulas for Sequences

Example

Find a formula for the general term a

in the sequence below, assuming that the pattern continues.

3 4 , − 4

8 , 5 16 , − 6

32 , 7 64 , − 8

128 , 9 256 , . . . i.e.

a

= 3

4 , a

= − 4

8 , a

= 5

16 , a

= − 6

32 , a

= − 7

64 , . . .

Limits of Sequences

Definition

If {a

} is a sequence that gets closer and closer to a number L as n gets large, then we say that the sequence converges and its limit is L and we write

n→∞

lim a

= L

If {a

} does not have a limit, then we say the sequence diverges. Example

n→∞

lim

1

n = 0

Limits of Sequences

Definition

If {a

} is a sequence that gets closer and closer to a number L as n gets large, then we say that the sequence converges and its limit is L and we write

n→∞

lim a

= L

If {a

} does not have a limit, then we say the sequence diverges.

Example

n→∞

lim

1

n = 0

Limits of Sequences

Definition

If {a

} is a sequence that gets closer and closer to a number L as n gets large, then we say that the sequence converges and its limit is L and we write

n→∞

lim a

= L

If {a

} does not have a limit, then we say the sequence diverges.

Example

n→∞

lim

1

n = 0

Finding Limits of Sequences

Example Find

n→∞

lim

3n

+ 1 n

+ n + 1 and

n→∞

lim

√ n ln n and

n→∞

lim (−1)

n

Table of Contents

Group Problem Solving

Question: Suppose whenever I take a bite out of a chocolate bar, I eat exactly half of the remaining chocolate bar. I will not eat the entire chocolate bar after finitely many bites, but will I eat the whole thing after infinitely many bites?

As a group, answer the above question, by finding answers to each of the following subquestions (there is a second set of subquestions on the next slide):

A.1. What proportion of the original chocolate bar do I eat on the first bite?

A.2. What proportion of the original chocolate bar do I eat on the second bite?

A.3. What proportion of the original chocolate bar do I eat on the third bite?

A.4. Let a

be the proportion of the original chocolate bar I eat on

the nth bite. Give a formula for a

.

Group Problem Solving

Question: Suppose whenever I take a bite out of a chocolate bar, I eat exactly half of the remaining chocolate bar. I will not eat the entire chocolate bar after finitely many bites, but will I eat the whole thing after infinitely many bites?

B.1. What proportion of the original chocolate bar have I eaten in total after the first bite?

B.2. What proportion of the original chocolate bar have I eaten in total after the second bite?

B.3. What proportion of the original chocolate bar have I eaten in total after the third bite?

B.4. Let s

= P

i =1

a

. What does s

represent? Based on the pattern so far guess a formula for s

.

B.5. What is lim

n→∞

s

? How does this answer the original question?

Geometric Sequences

A sequence {a

} of the form

a, ar , ar

, ar

, ar

, . . . i.e. a

=

ar

, is called a geometric sequence. r is called the geometric ratio.

Example

Find a and r for the geometric sequence below 3

2 , −6, 24, −96, 384, . . .

Geometric Sequences

A sequence {a

} of the form

a, ar , ar

, ar

, ar

, . . .

i.e. a

= ar

, is called a geometric sequence. r is called the geometric ratio.

The N’th partial sum of the geometric sequence is s

=

N

X

n=1

a

=

N

X

n=1

ar

= a + ar + ar

+ · · · + ar

Let’s find a formula for s

by considering rs

: rs

=

r

N

X

n=1

a

= r

N

X

i =n

ar

= ar + ar

+ · · · + ar

+ ar

s

− rs

=

a − ar

s

=

a(1 − r

)

1 − r

Geometric Sequences

A sequence {a

} of the form

a, ar , ar

, ar

, ar

, . . . i.e. a

=

ar

, is called a geometric sequence. r is called the geometric ratio. The N’th partial sum of the geometric sequence is

s

= a(1 − r

) 1 − r Example

Find the 10’th partial sum for the geometric sequence below 3

2 , −6, 24, −96, 384, . . .

Geometric Sequences

A sequence {a

} of the form

a, ar , ar

, ar

, ar

, . . .

i.e. a

= ar

, is called a geometric sequence. r is called the geometric ratio.

The N’th partial sum of the geometric sequence is

s

=

N

X

n=1

ar

= a + ar + ar

+ · · · + ar

s

= a(1 − r

) 1 − r

N→∞

lim s

= (

a

1−r

if |r | < 1

DNE if |r | ≥ 1

Geometric Sequences

The N’th partial sum of the geometric sequence is

s

=

N

X

n=1

ar

= a + ar + ar

+ · · · + ar

= a(1 − r

) 1 − r

N→∞

lim s

= (

1−r

if |r | < 1 DNE if |r | ≥ 1 Definition

A geometric series is an infinite sum

∞

X

n=1

ar

= lim

N→∞

s

The geometric series converges if

|r | < 1

and diverges if

|r | ≥ 1

.

Geometric Sequences

Definition

A geometric series is an infinite sum

∞

X

n=1

ar

= lim

N→∞

s

= lim

N→∞

N

X

n=1

ar

The geometric series converges if |r | < 1 and diverges if |r | ≥ 1.

Example

The geometric series representing the problem we started with today

is

Geometric Sequences

Definition

A geometric series is an infinite sum

∞

X

n=1

ar

= lim

N→∞

s

The geometric series converges if |r | < 1 and diverges if |r | ≥ 1.

Example

Write down the geometric series for the geometric sequence below and find it converges or diverges.

3

2 , −6, 24, −96, 384, . . .

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Partial Sums and Series

Consider a general sequence {a

}, i.e.

a

, a

, a

, a

, . . .

The N’th partial sum of the sequence is given by s

=

a

+ a

+ · · · + a

=

N

X

n=1

a

Notice that we get a new sequence {s

}.

The series of the sequence is the limit of the sequence s

, which we can think of as the infinite sum

∞

X

n=1

a

= a

+ a

+ a

+ · · · = lim

N→∞

s

which may or may not exist. If the limit exists, we say the series

P

a

converges and if the limit does not exist, we say it

diverges.

Partial Sums and Series

The N’th partial sum of the sequence is given by s

= a

+ a

+ · · · + a

=

n

X

n=1

a

Example

Let a

= 2n − 1. Find the first 4 partial sums. Guess a formula for

s

based on a pattern. (What does this remind you of?)

Telescoping Series Example

Compute

∞

X

n=1

 1 n − 1

n + 1



Divergence Test

Theorem

If a sequence {a

} does not converge to 0, then its series diverges, i.e.

n→∞

lim a

6= 0 =⇒

∞

X

n=1

a

diverges.

Remark

It is NOT true that

n→∞

lim a

= 0 =⇒

∞

X

n=1

a

converges.

Table of Contents

Divergence Test

Theorem

If a sequence {a

} does not converge to 0, then its series diverges, i.e.

n→∞

lim a

6= 0 =⇒

∞

X

n=1

a

diverges.

Example

What does the divergence test say about

∞

X

n=1

1 n = 1

1 + 1 2 + 1

3 + · · ·?

Table of Contents

P ∞ n=1

1 n

Example

Determine whether or not

∞

X

n=1

1 n

= 1

1

+ 1 2

+ 1

3

+ · · ·

converges.

Group Problem Solving

Instructions: Following the previous example, determine whether or not P

n=11

n

converges by doing the following steps on the Jamboard:

1. Graph the function f (x ) =

and plot the sequence as points on the graph.

2. Draw a Riemann sum that is in fact equal to the series, but is an overestimate of the integral R

1 1 x

dx .

3. Using Step 2., determine the convergence of P

n

.

Do a similar procedure to the above to determine the convergence of P

n=1 √1 n

.

As a group, guess what the general rule is:

∞

X

n=1

1 n

=

( converges for

diverges for

The Integral Test

Theorem (The Integral Test)

Suppose the sequence a

is given by f (n) where f is a continuous, positive, decreasing sequence on [1, ∞). Then

∞

X

n=1

a

converges if and only if Z

1

f (x )dx converges.

Example

Determine the convergence of the series of the sequence

e

, 2e

, 3e

, . . .

Table of Contents

The (Basic) Comparison Test

Theorem (The (Basic) Comaprison Test)

Suppose {a

} and {b

} are two sequences such that 0 ≤ a

≤ b

for all n. Then

0 ≤

∞

X

n=1

a

≤

∞

X

n=1

b

In particular, If

P

a

converges, then

P

b

converges also.

If

P

b

diverges, then

P

a

diverges also.

(What does this remind you of?)

The Limit Comparison Test

Theorem (The Limit Comaprison Test)

Suppose {a

} and {b

} are two sequences such that

n→∞

lim a

b

= L where L is a finite number and L 6= 0.

Then

∞

X

n=1

a

converges if and only if

∞

X

n=1

b

converges.

Group Problem Solving

Instructions: For each pair of series below, determine the

convergence of one of them without using a comparison test. Then use a comparison test with these two series to determine the convergence of the other.

∞

X

n=1

1 2

∞

X

n=1

1 2

+ 1

∞

X

n=1

1 2

∞

X

n=1

1 2

− 1

∞

X

n=1

1 n

∞

X

n=1

n

+ 1 n

− n

+ 1

∞

X

n=1

1 n

∞

X

i =1

√

i

+ 1(i

+ 3)

i

− i

+ i

Group Problem Solving

Instructions: For each pair of series below, determine the

convergence of one of them without using a comparison test. Then use a comparison test with these two series to determine the convergence of the other.

∞

X

n=1

sin n + 2 n

∞

X

n=1

1 n

∞

X

n=1

n sin

n + cos n + 3 n

+ 5 − 4 cos n

∞

X

n=1

1 n

∞

X

n=1

e

∞

X

n=1

12e

log(n + 1)

∞

X

n=1

e

n

∞

X

N=1

1

N