Sequences and Series Flashcards
Sequence
Definition
a list (usually infinite) of numbers indexed by integers (usually ℕ)
Infinite Sequence
A More Precise Definition
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
Bounded Below
Definition
a sequence is bounded below if and only if there exists some c∈ℝ such that an>=c for all n∈ℕ
Bounded Above
Definition
a sequence is bounded above if and only if there exists some c∈ℝ such that an<=c for n∈ℕ
Unbounded
Definition
a sequence is unbounded if and only if there does not exist c∈ℝ such that |an|<=c for all n∈ℕ
Bounded
Definition
a sequence is bounded if it is bounded above and below
Increasing Sequence
Definition
a sequence is increasing if a(n+1) >= an
Decreasing Sequence
Definition
a sequence is decreasing if a(n+1) <= an
Strictly Increasing Sequence
Definition
a sequence is strictly increasing if a(n+1) > an
Strictly Decreasing Sequence
Definition
a sequence is strictly decreasing if a(n+1) < an
Convergence
Definition
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
Divergence
Definition
If a sequence does not converge, it is said to diverge
Monotonic
Definition
a sequence that either never decrease or never decreases
Limit Comparison
Assume an and bn satisfy an >= bn for all n>N0∈ℕ
Assuming an -> a and bn -> b, a>=b
If bn -> ∞, then an -> ∞
Adding Sequences and Limits
let an -> a and bn -> b
an + bn -> (a+b)
Sandwich / Squeeze Rule
Assume an and cn satisfy an -> l and cn -> l
If an<= bn <= cn, for all n>No∈ℕ
Then bn -> l as well
Subsequence
Definition
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 -> ±∞
What is the limit of an = 1/n^c, c>0
limit = 0
What is the limit of an = c^n
for c>1, limit = ∞
for c=1, limit = 1
for |c| < 1, limit = 0
for c <= -1, the sequence diverges, no limit exists
What is the limit of an = n^c / d^n
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)
Bounded and Convergent Sequences
- a bounded sequence need not be convergent
- but all convergent sequences are bounded
Monotone Convergence Theorem
let an be a sequence that is either decreasing or increasing & bounded below (decreasing) or above (increasing), then the sequence converges
Infinite Series
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
Convergent Series
Definition
A series Σ Sn is convergent if the sequence Sn is convergent
Divergent Series
Definition
A series Σ Sn diverges if it does not converge
The series diverges to ±∞ if the sequence Sn diverges to ±∞
Harmonic Series, ak = 1/k
ak = 1/k, (k=1->∞) Σ ak
diverges to infinity
Vanishing Test
If (n->∞)lim an = 0 Then (n=1->∞) Σ an converges
Harmonic Series ak = 1 / k^α
(k=1->∞) Σ ak
converges for α>1
diverges for α<=1
Geometric Series ak = A^k
(k=1->∞) Σ ak
converges to 1/(1-A) for |A| < 1
Comparison Test
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 < +∞
Ratio Test
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
Arithmetics of Convergent Series
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
Alternating Series
Definition
A series Σ ak is alternating if and only if,
ak * ak+1 < 0 for all k∈ℕ
The Leibniz Test
Any alternating series Σ ak satisfying (k->∞)lim ak = 0
and |ak+1| < |ak| for all n>N0∈ℕ
is convergent
Does Σ (-1)^k / k^α converge?
converges for α>0
Power Series
Definition and Convergence
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)
