JB Flashcards
Use sign analysis to find the solution set for x^2 > 9
Re arrange, to =0. Then make a table of f(x) and factors of f(x) (LHS) and on the top have important values of x.
Then depending on whether f(x) is =ve,-ve or 0 and what the equality is you’ll have your solution set.
|xy| =?
|x|.|y|
What’s the triangle inequality?
For any a,b ∈ R,
|a+b| ≤ |a|+|b|.
Prove the triangle inequality
For a,b ∈ R, ab≤|ab|, so that ab≤|a||b| Thus a^2 + 2ab + b^2 ≤ a^2 + 2|a||b| + b^2 = |a|^2 + 2|a||b| + |b|^2 Factored = |a+b|^2 ≤ (|a|+|b|)^2 Taking the positive square root |a+b| ≤ |a|+|b| QED
What’s the definition for a sequence of real numbers tending to infinity?
Given any real number A>0 there exists N∈Nat such that an > A for all n>N.
or
∀A > 0 ∃N ∈ N s.t. an > A ∀n > N.
Prove that the sequence (an) given by an = √n tends to infinity.
Let A > 0 be given. We are required to find an N ∈ Nat s.t √n > A for all n > N.
Notice that if n > A^2 then √n > A. This suggests choosing N to be any natural
number greater than or equal to A^2. With this choice of N we have
n > N =⇒ n > A^2 =⇒ √n > A,
as required.
Prove that the sequence (an), given by
an = n^2 + n/(n + 5)
tends to infinity.
(†) an = n2 + n/(n + 5) ≥ n^2/(n + 5) ≥n^2/(n + 5n) = n/6
for all n ∈ Nat.
Let A > 0 be given, and choose N to be any nat ≥
6A. With this choice of N we have
n > N =⇒ n > 6A =⇒ n/6 > A =⇒ an > A,
as required. Here in the last implication we have used (†).
Thm 1.3 - Suppose N0 ∈ N and that (an) and (bn) are sequences of real numbers satisfying
an ≥ bn for all n ≥ N0. ?
If bn → inf, what can we conclude?
an → inf
What’s the proof for Thm 1.3 (if an>bn and b->inf)?
Let A > 0. Since bn → inf there exists N′ ∈ N such that
bn > A for all n > N′.
Now let N = max{N′,N0} and observe that by the hypotheses of the theorem,
an ≥ bn > A for all n > N.
Hence an → inf.
What is the null sequence test?
If a series ∑(inf,n=1)an converges then an->0 as n->0
What is the Comparison test?
Suppose that 0≤an≤bn ∀ n∈ |N.
(1) If ∑(inf,n=1)bn converges then ∑(inf,n=1)an converges.
(2) If ∑(inf,n=1) an diverges then ∑(inf,n=1)bn diverges.
What is the Ratio Test?
Let (an) be a sequence of non-negative real numbers s.t
lim(n->inf) a(n+1)/an = r, then the series ∑(inf,n=1)an
(1) converges if r<1
(2) diverges if r>1
if r=1, the ratio test tells us nothing
What is the Integral test?
Let f: |R->|R be cts,decreasing, and positive on[0,inf). Then ∑(inf,n=1) f(n) converges iff the sequence of integrals ∫(n,1) f(x)dx converges as n->inf
What is the Alternating Series Test?
Suppose the non-negative sequence (an)
(i) is decreasing,
(ii) and converges to 0.
Then a1 - a2 +a3 - a4 +… = ∑(inf,n=1) (-1)^(n+1) an converges.
What is the absolute convergence test?
If ∑(inf,n=1)an is absolutely convergent then ∑(inf,n=1)an is convergent in the usual sense.
What is the definition of a increasing sequence of real numbers?
if a(n+1)≥a(n) ∀ n∈ |N.
What is the definition of a decreasing sequence of real numbers?
if a(n+1) ≤ a(n) ∀ n∈ |N.
What is the definition of a strictly increasing sequence of real numbers?
if a(n+1) > a(n) ∀ n∈ |N.
What is the definition of a strictly decreasing sequence of real numbers?
if a(n+1) < a(n) ∀ n∈ |N.
State MCT
(i) If (an) is increasing and bounded above, then (an) converges.
(ii) If (an) is decreasing and bounded below, then (an) converges.
What does it mean for a sequence to be bounded from below?
an is bounded below if there exists some M ∈ |R such that an ≥ M for all n ∈ N.
What does it mean for a sequence to be bounded from above and below?
(an) is bounded if it is both bounded above and below; i.e. there exist M1,M2 ∈ |R such
that M1 ≤ an ≤ M2 for all n ∈ N.
