Sequences and Series

Question 1

Sequence

Definition

Answer 1

a list (usually infinite) of numbers indexed by integers (usually ℕ)

Question 2

Infinite Sequence

A More Precise Definition

Answer 2

a prescription associating each natural number to a unique real number, such a map:
M : N -> R, n|—>an
is a special case of the general notion of a function

Question 3

Bounded Below

Definition

Answer 3

a sequence is bounded below if and only if there exists some c∈ℝ such that an>=c for all n∈ℕ

Question 4

Bounded Above

Definition

Answer 4

a sequence is bounded above if and only if there exists some c∈ℝ such that an<=c for n∈ℕ

Question 5

Unbounded

Definition

Answer 5

a sequence is unbounded if and only if there does not exist c∈ℝ such that |an|<=c for all n∈ℕ

Question 6

Bounded

Definition

Answer 6

a sequence is bounded if it is bounded above and below

Question 7

Increasing Sequence

Definition

Answer 7

a sequence is increasing if a(n+1) >= an

Question 8

Decreasing Sequence

Definition

Answer 8

a sequence is decreasing if a(n+1) <= an

Question 9

Strictly Increasing Sequence

Definition

Answer 9

a sequence is strictly increasing if a(n+1) > an

Question 10

Strictly Decreasing Sequence

Definition

Answer 10

a sequence is strictly decreasing if a(n+1) < an

Question 11

Convergence

Definition

Answer 11

A sequence id convergent if it approaches some limit

A sequence Sn has a limit S if, for any ɛ>0, there exists an n>N such that |Sn-S| < ɛ

Question 12

Divergence

Definition

Answer 12

If a sequence does not converge, it is said to diverge

Question 13

Monotonic

Definition

Answer 13

a sequence that either never decrease or never decreases

Question 14

Limit Comparison

Answer 14

Assume an and bn satisfy an >= bn for all n>N0∈ℕ
Assuming an -> a and bn -> b, a>=b
If bn -> ∞, then an -> ∞

Question 15

Adding Sequences and Limits

Answer 15

let an -> a and bn -> b

an + bn -> (a+b)

Question 16

Sandwich / Squeeze Rule

Answer 16

Assume an and cn satisfy an -> l and cn -> l
If an<= bn <= cn, for all n>No∈ℕ
Then bn -> l as well

Question 17

Subsequence

Definition

Answer 17

a subsequence of a sequence (an) is a sequence that arises by omitting numbers from (an). It can be written as (bn) = (akn)
If an -> l, then bn -> l
Or if an -> ±∞ then bn -> ±∞

Question 18

What is the limit of an = 1/n^c, c>0

Answer 18

limit = 0

Question 19

What is the limit of an = c^n

Answer 19

for c>1, limit = ∞
for c=1, limit = 1
for |c| < 1, limit = 0
for c <= -1, the sequence diverges, no limit exists

Question 20

What is the limit of an = n^c / d^n

Answer 20

Both n^c and d^n tend to ∞

But in this case an has limit 0 as exponential growth (d^n) beats polynomial growth (n^c)

Question 21

Bounded and Convergent Sequences

Answer 21

• a bounded sequence need not be convergent

- but all convergent sequences are bounded

Question 22

Monotone Convergence Theorem

Answer 22

let an be a sequence that is either decreasing or increasing & bounded below (decreasing) or above (increasing), then the sequence converges

Question 23

Infinite Series

Answer 23

Given a sequence (an), there is an associated infinite series, (n=1->∞)Σ an OR (k=1->n)Σ an also known as the nth partial sum

Question 24

Convergent Series

Definition

Answer 24

A series Σ Sn is convergent if the sequence Sn is convergent

Question 25

Divergent Series

Definition

Answer 25

A series Σ Sn diverges if it does not converge

The series diverges to ±∞ if the sequence Sn diverges to ±∞

Question 26

Harmonic Series, ak = 1/k

Answer 26

ak = 1/k, (k=1->∞) Σ ak

diverges to infinity

Question 27

Vanishing Test

Answer 27

If (n->∞)lim an = 0 Then (n=1->∞) Σ an converges

Question 28

Harmonic Series ak = 1 / k^α

Answer 28

(k=1->∞) Σ ak
converges for α>1
diverges for α<=1

Question 29

Geometric Series ak = A^k

Answer 29

(k=1->∞) Σ ak

converges to 1/(1-A) for |A| < 1

Question 30

Comparison Test

Answer 30

IF (k=1->∞) Σ bk < +∞ such that bk>=0 for all k>=1 and
(k=1->∞) Σ ak. Then, if there exists a constant c >= 0
such that |ak| < c*bk for all k, then (k=1->∞) Σ ak < +∞

Question 31

Ratio Test

Answer 31

Consider (k=1->∞) Σ ak
and define, L = (k->∞) lim |ak+1| / |ak|
If L<1 the series converges absolutely
If L>1 the series diverges
If L=1 we have no information on the series

Question 32

Arithmetics of Convergent Series

Answer 32

If two series (k=1->∞) Σ ak and (k=1->∞) Σ bk both converge, then
(k=1->∞) Σ ak + bk = Σ ak +Σ bk < +∞
(k=1->∞) Σ α*ak = α Σ ak < +∞, for all α∈ℝ
If the two series Σ ak and Σ bk are absolutely convergent then their product is convergent

Question 33

Alternating Series

Definition

Answer 33

A series Σ ak is alternating if and only if,

ak * ak+1 < 0 for all k∈ℕ

Question 34

The Leibniz Test

Answer 34

Any alternating series Σ ak satisfying (k->∞)lim ak = 0
and |ak+1| < |ak| for all n>N0∈ℕ
is convergent

Question 35

Does Σ (-1)^k / k^α converge?

Answer 35

converges for α>0

Question 36

Power Series

Definition and Convergence

Answer 36

An infinite series in the form Σ ck*x^k for some constants ck
Its radius of convergence ir R = (k->∞)lim |ck| / |ck+1|
It converges for x ∈ (-ℝ,ℝ)