Topic 5: Sequences and Series

Question 1

What is the definition of a sequence (2)

Answer 1

-A sequence is an infinite list of real numbers
-Equivalently, we can view this as function f: ℕ -> ℝ

Question 2

What is the definition of a sequence converging to a limit (4)

Answer 2

-Let (an) be a sequence, and suppose L ∈ ℝ is some finite real number
-Then an tends/converges to the limit L, if for any ε > 0, there exists N ∈ ℕ such that |an - L| < ε for all n > N
-an tends/converges to any Y value if for a very small number ε (a Y value error number), there exists an x value N such that the absolute distance between the sequence and the limit is less than the y error value for all x values greater than N
-We write lim(n -> ∞) an = L or an -> L as n -> ∞

Question 3

How can we prove an = (2n+3)/(3n-4) -> 2/3 (7)

Answer 3

-Let an = (2n+3)/(3n-4), set L = 2/3
-Suppose |an - L| = |(2n+3)/(3n-4) - 2/3| < ε
-|17/(9n-12)| < ε
-For n>1, 17/(9n-12) < ε => n > (17+12ε)/9ε
-so if n > (17+12ε)/pε, |an - 2/3| < ε
-Set N = [(17+12ε/9ε]
-Therefore, for n > N, |an - 2/3| < ε

Question 4

How can we define a sequence tending to ∞/-∞ (2)

Answer 4

-A sequence (a_n) tends to ∞ if, given any A>0, there exists some N = N(A) ∈ ℕ such that a_n > A for all n > N (for any Y value A there exists some x value a where at that point you have Y value A, and the y value of the sequence is greater than A for all x values greater than N)
-Similarly, a_n -> -∞ if for A < 0, there exists N = N(A) ∈ ℕ such that a_n < A for all n > N

Question 5

How can we propose one sequence tends to 0 if we know another one does (2)

Answer 5

-Let (a_n) and (b_n) be sequences, and suppose there exists some N ∈ ℕ such that a_n > b_n for all n > N.
-Then, if b_n -> ∞ as n -> ∞, we know a_n -> ∞

Question 6

How can we prove one sequence tends to 0 if we know another one does ()

Answer 6

-Let A>0
-Since b_n -> ∞, there exists N₁ ∈ ℕ such that b_n > A for all n > N₁
-Now choose N = max {N₀, N₁}, where N₀ is the point where a_n > b_n
-So for n > N we have a_n > b_n > A, so a_n also -> ∞ as n -> ∞

Question 7

What is the definition of a convergent vs divergent sequence (2)

Answer 7

-A sequence (a_n) is convergent if it tends to a finite limit as n -> infinity
-A sequence is divergent if it tends to +- infinity as n -> infinity

Question 8

What is the definition of a bounded sequence (2)

Answer 8

-A sequence (a_n) is bounded if there exists some finite M > 0 such that |a_n| ≤ M for all n ∈ ℕ
-A sequence is bounded below if there exists M_ ∈ ℝ such that a_n ≥ M_ for all n ∈ ℕ, and bounded above if opposite

Question 9

Are all bounded sequences convergent (1)

Answer 9

-No, not all bounded sequences are convergent (sin is bounded between +- 1 but doesn’t converge)

Question 10

How can we prove that convergent sequences are bounded (1,6)

Answer 10

-Proposition: convergent sequences are bounded

Proof:
-Suppose (a_n converges to finite limit L as n -> ∞
-Then there exists some N ∈ ℕ such that for any ε > 0, we have |a_n - L| < ε for n > N
-In this case, chose ε = 1
-Rearranging this and using the triangle inequality gives us |a_n| < L + 1 for all n > N
-Now let M = max {|a_n, |L + 1|}
-Then for all n ∈ ℕ, we have |a_n| ≤ M

Question 11

What are 2 examples of the algebra of limits (2)

Answer 11

Suppose a_n -> a and b_n -> b as n -> ∞

-(a_n + b_n) -> (a + b) as n -> ∞
-(a_n)(b_n) -> (ab) as n -> ∞

Question 12

What is the sandwich rule (2)

Answer 12

Theorem:
-Suppose a_n ≤ b_n ≤ c_n for all n > N ∈ ℕ
-if a_n -> L, c_n -> L as n -> ∞, b_n -> L

Question 13

How can we prove the sandwich rule (FINISH FLASHCARD WITH PAPER ()

Answer 13

-Let ε > 0
-Since a_n -> L, as n -> ∞, there exists some N₁ ∈ ℕ such that |a_n - L| < ε for n > N
-Similarly, there exists some n

Question 14

What is an increasing/decreasing sequence (2,2)

Answer 14

-A sequence is increasing if a_n+1 ≥ a_n for all n ∈ ℕ
-A sequence is strictly increasing if a_n+1 > a_n for all n ∈ ℕ

-A sequence is decreasing if a_n+1 ≤ a_n for all n ∈ ℕ
-A sequence is strictly decreasing if a_n+1 <
a_n for all n ∈ ℕ

Question 15

What is a monotonic sequence (2)

Answer 15

-A sequence is monotonic if it is either increasing or decreasing for n ∈ ℕ
-A sequence is strictly monotonic if it is either strictly increasing or decreasing for n ∈ ℕ

Question 16

What happens to the upper limit of an increasing sequence, and what theorem can we get as a result of this (2,1,1)

Answer 16

If a sequence is increasing, it is either
-Bounded above
-Unbounded

-Corollary: an increasing sequence converges if and only if it is bounded above

-Theorem: A bounded monotonic sequence converges

Question 17

What is an example of a subsequence (2)

Answer 17

-Suppose a_n = 1/(2ⁿ) = 1/2, 1/4, …
-If we delete every even-numbered entry, we get a’_n = 1/(2^2n-1)

Question 18

What is the definition of a subsequence (1,3)

Answer 18

-A subsequence is a sequence that can be derived from another sequence by deleting some/no elements without changing the order of the remaining elements

-Suppose (a_n)^∞_n=1 is a sequence
-Let (n_k)^∞_k=1 be a strictly increasing sequence of positive integers
-Then (a_nk)^∞_k=1 = a_n1 > a_n2 is a subsequence of (a_n)^∞_n=1

Question 19

What is the relationship between the limits of sequences and subsequences (2)

Answer 19

-Suppose (a_n) is a convergent sequence where a_n -> L as n -> ∞
-Then any subsequence of (a_n) also converges to L as n -> ∞

Question 20

How do we prove that every sequence has a monotonic subsequence (3, 2, 4,1)

Answer 20

-We say element a_p of sequence a_n is a terrace point if a_p ≥ a_n for all n ≥ p (kind of like a max point for a sequence)
-Every point after a terrace point is lower than it
-The sequence either has infinitely many terrace points, or finitely many (or 0)

-If a_n has infinitely many terrace point, we can label these as a_n1, a_n2…, where n₁ < n₂
-So a_n1, a_n2… is a decreasing subsequence of a_n

-If a_n has finitely many (or 0) terrace points, find n₁ such that all the terrace points a_p occur for p < n₁
-Since n₁ isn’t a terrace point, there must be some n₂ > n₁ with a_n2 > a_n1
-Since n₂ also isn’t a terrace point, there must be some n₃ > n₂ with a_n3 > a_n2
-We can then continue in this way to show the existence of infinitely many n₁, n₂ …. such that a_n1 < a_n2 …, which is a strictly increasing subsequence

-Therefore, a_n contains either a decreasing or strictly increasing subsequence, which is therefore monotonic

Question 21

What is the Bolzano-Weierstrass theorem (1)

Answer 21

-Every bounded sequence has a convergent subsequence

Question 22

How can we write the sum of the first n terms of g_k = ar^k-1 (3)

Answer 22

-Suppose a, r ∈ ℝ, with r not equal to 1
-Consider (g_r)_k=1^∞ defined by g_k = ar^k-1
-S_n = ∑g_k = a((1-r^a)/(1-r))

Question 23

How can we write the sum of the first n terms of g_k = ar^k-1 if |r| < 1 (3)

Answer 23

-If |r| < 1, rⁿ -> 0 as n -> ∞
-Thus, ∑ar^k-1 = a/1-r
-This is the geometric series

Question 24

What is the definition of a series (2)

Answer 24

-A series is the sum of the (infinitely many) terms in a sequence
-We write this as ∑_k=1^∞ a_n, given sequence a_n

Question 25

What is the definition of the partial sum, and how do we use this to find if the infinite series converges or diverges (2,2,1)

Answer 25

-Let a_n be a sequence
-The partial sum of the first K terms = ∑_n=1^k a_n

-The infinite series ∑_k=1^∞ a_k converges is the partial sum (s_k) converges to a finite limit as k -> ∞.
-We call this limit ‘S’ the sum of the series and write ∑_k=1^∞ a_k = S

-If no such limit exists, we say the series diverges

Question 26

What is the sum rule for the convergence of 2 series (1)

Answer 26

-If the series’ ∑_k=1^∞ a_k and ∑_k=1^∞ b_k both converge, do does ∑_k=1^∞ a_k + b_k

Question 27

What are some tests for convergence (5)

Answer 27

-Sum rule
-Null sequence test
-Comparison test
-Ratio test
-Alternating series test

Question 28

What is the null sequence test for the convergence of series (1,2)

Answer 28

-The series ∑_k=1^∞ a_k only converges if a_k -> 0 as k -> 2

-This is a necessary but not sufficient condition for the convergence of a series
-For example, 1/k -> 0 as k -> ∞, but ∑_k=1^∞ 1/k still doesn’t converge

Question 29

What is the comparison test for the convergence for series (1,3,1)

Answer 29

-Let ∑_k=1^∞ a_k and ∑_k=1^∞ b_k be series

If:
-a_k, b_k ≥ 0 for all k ∈ ℕ
-a_k ≤ Mb_k for all k ∈ ℕ and M > 0
-∑_k=1^∞ b_k converges

-Then, ∑_k=1^∞ a_k converges

Question 30

What is the comparison test for the divergence for series (1,3,1)

Answer 30

-Let ∑_k=1^∞ a_k and ∑_k=1^∞ b_k be series

If:
-a_k, b_k ≥ 0 for all k ∈ ℕ
-a_k ≥ Mb_k for all k ∈ ℕ and M > 0
-∑_k=1^∞ b_k diverges

-Then, ∑_k=1^∞ a_k diverges

Question 31

How can we prove that ∑_k=1^∞ 1/k doesn’t converge (4)

Answer 31

-Let S = ∑_k=1^∞ 1/k = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …
-(1) = 1 = b₁, (1/2) = 1/2 = b₂, (1/3 + 1/4) > 1/2 = b₃, (1/5 + 1/6 + 1/7 + 1/8) > 1/2 = b₄
-This series is bounded below by the series ∑_k=1^∞ 1/2
-Since ∑_k=1^∞ 1/2 diverges by the null sequences test, therefore so does ∑_k=1^∞ 1/k

Question 32

What is the ratio test for the convergence of series (2,2)

Answer 32

-Take a series ∑_k=1^∞ 1
-Say (|a_k+1/a_k|) ->L
-If L < 1, ∑_k=1^∞ a_k converges
-If L > 1, ∑_k=1^∞ a_k diverges

Question 33

What is the alternating series test for the convergence of series (3,1)

Answer 33

For sequence a_k, If:
-a_k > 0 for all k ∈ ℕ
-a_k is decreasing
-a_k -> 0

-Then ∑_k=1^∞
(-1)^k a_k

Question 34

How can we prove series ∑_k=1^∞ (-1)^k/k converges (1,3,1)

Answer 34

-Consider ∑_k=1^∞ 1/k, which doesn’t converge

-However, 1/k > 0 for all k ∈ ℕ
-1/k -> 0 as k -> infinity
-This is thus a decreasing dequence

-Then, by the alternating series test, series ∑_k=1^∞ (-1)^k/k converges