Sequences (Chapter 7)

Question 1

Explain why

Answer 1

Question 2

What is euler’s series?

Does it diverge or converge?

Answer 2

It converges to pi²/6

Also known as basel series

Question 3

What is the harmonic series?

Does it diverge or converge?

Answer 3

It diverges

Question 4

What is grandi’s series

Answer 4

Question 5

Answer 5

Question 6

What is the name of this series

Answer 6

It is a telescopic series

Question 7

True or false

All telescopic series converges

Answer 7

False

Question 8

What is the general form for any p series?

Answer 8

For any real number p

Question 9

When does a p series converge and when does it diverge?

Answer 9

Question 10

What is the if statement related to the divergence test

Answer 10

Question 11

Answer 11

Question 12

What do all p series have in common (in regards to the first term)

Answer 12

The first term is always 1.

1/n^p = 1 for n=1 everytime (no matter the p)

Question 13

True or false?

If the tail of a series diverges, then the whole series also diverges

Answer 13

True

If you know the tail diverges, the sum will go to infinity

Question 14

True or false

If the tail of a series converges, then the whole series also converges

Answer 14

True

If you know the tail converges, then you just add the head to the converging tail and it will become a number (rather than infinity)

Question 15

Identify if the following is a p series or not

Answer 15

1/1+1/2+1/3+1/4+1/5+…+1/n

It is a p-series

Question 16

Idenitfy the name of this series. Identify when it converges (and how to calculate what it converges to) and when it diverges

Answer 16

Note that a is the first term and r is the common ratio

Question 17

Answer 17

Question 18

Explain the comparison test

Answer 18

Question 19

Answer 19

Question 20

Answer 20

Question 21

Revision

Question 22

What is defined as an alternating series

Answer 22

When the sign of each term alternates between + and -

(+,-,+,-,+,-)

Question 23

True or false?

The alternating harmonic series is convergent

Answer 23

True

Question 24

Fill the box

Answer 24

The alternating series test can only be used to show convergence, not divergence.

Question 25

Answer 25

Question 26

Answer 26

Question 27

When do we say a series converges absolutely?

Answer 27

Question 28

When do we say a series converges condititionally

Answer 28

| Basically when a series converges but not necessarily absolutely

Question 29

Answer 29

Question 30

# Is the following series conditionally or absolutely convergent?

Answer 30

## Footnote You don't actually need to show convergence of the orginal series. Absolute convergence implies convergence

Question 31

# Is the following series conditionally or absolutely convergent?

Answer 31

Question 32

# True or false? If a series is absolutely convergent, then it is convergent

Answer 32

True

Question 33

Define the ratio test

Answer 33

If each term in the series gets less and less, then it will absolutely converge. If each term in the series gets more and more, then it will diverge

Question 34

Answer 34