Chapter 8 sequence sand series Flashcards
Definition: pointwise convergence
We say that the sequence (f_n) converges pointwise to a function f:[a,b] TO R
If for each t in [a,b] we have
Limit as n tends to infinity of f_n(t) =f(t)
Ie if the sequence converges pointwise to f for each of those ts and ε bigger than 0 there exists N∈N st |f_n (t) - f(t)| is less than ε whenever n is bigger than or equal to N.
N depends on both t and ε!!!
Definition uniform convergence:
Let f_n : [a,b] to R.
We say that the sequence (f_n) converges uniformly to a function f:[a,b] to R
When for all ε bigger than 0 we have N∈N st |f_n (t) - f(t)| is l
Less than
ε for all n bigger than or equal to N and t in [a,b]
N DOESNT depend on t
Only on ε!!!
Uniform convergence of a sequence to a function implies pointwise convergence of the same function- stronger!
Proposition 8.1.6: equivalent statements for uniform convergence and supremum
Consider a sequence of functions f_n:[a,b] to R.
Let f:[a,b] to R.
Then the following are equivalent:
• the sequence (f_n) converges uniformly to f
•let M_n = sup{|f_n (t) -f(t)| t∈[a,b]}. Then M_n tends to 0 as n tends to infinity
This conditions on the M_n s is the easiest way to prove uniform convergence
Theorem: uniform limit theorem
Let f_n : [a,b] to R be continuous for each n∈N. suppose the sequence (f_n) converges uniformly to a function f:[a,b] to R. Then f is continuous.
Proof I
Theorem 8.3.1: uniform convergence theorem
Let f_n: [a,b] to R be a CONTINUOUS FUNCTION for n∈N.
Suppose the sequence (f_n) converges uniformly to a function f.
Then (limit as n tends to infinity Of integral over [a,b] of f_n (t) .dt) = (integral over [a,b] of f(t) .dt)
This allows us to swap limits and integral signs
Corollary to uniform convergence theorem 8,3,1
Consider differentiable functions f_n :[a,b] to R.
Suppose (f_n) converges pointwise to a function f and the sequence of derivatives (f_n ‘) converges uniformly to a function g. Then f is differentiable and f’=g.
Thus under suitable conditions
We can swap limits and differentiation.
Definition: uniformly continuous
Let f:[a,b ] to R. We say f is uniformly continuous on [a,b] if for all
εbigger than 0 we have δ bigger than 0 such that if |x-y| is less than δ then | f(x) -f(y)| is less than ε for all x,y ∈[a,b]
—–
Delta depends on epsilon only not on chosen point . For a given epsilon the same delta has to work across while interval.
If F is uniformly continuous then f is continuous
True or false
For f :R to R
tRUE
If f is continuous then f is uniformly continuous
f: R to R
FALSE
example.
Only if on [a,b]
f: R to R
f(x) = x^2 is continuous but not uniformly continuous
Ie for x and y in reals with |x-y| = c bigger than 0
|f(x) -f(y)| = |x^2 -y^2| = |(x-y)(x+y)| = c|x+y|
So if x,y bigger or equal to 1/c then This is bigger than 1. Taking epsilon =1 there is no delta in delta epsilon condition for convergence. Thus f is not uniformly continuous.
Theorem 8.4.3 continuous bs uniformly continuous
Let f: [a,b] to R be continuous then f is uniformly continuous
Only true for [a,b]
Theorem 8.4.4
Riemann integrabke and continuity
Let f:[a,b] to R be continuous
Then f is Riemann integrable
Definition 8.5.1: partial sums and convergence pointwise and uniformly
Let (f_n) be a sequence of functions f_n :[a,b] to R
We consider the sequence of partial sums (s_n) where s_n:[a,b] to R is defined by
s_n (t) = f₁(t) + f₂(t) +…+ f_n (t).
We say that the series ( Σ from n=1 to infinity of f_n)
Converges pointwise if
For each t ∈[a,b] the series sum of f_n(t) converges
Uniformly summable or converges uniformly of sequence of the partial sums converges uniformly
Theorem 8.5.2 uniformly summable sequence and continuity
Let (f_n) be a uniformly summabke sequence of continuous functions f_n :[a,b] to R.
Then the function f:[a,b] to R is given by
f(t) =
Σ from n=1 to infinity of
f_n(t)
(exists) and is continuous
Theorem 8.5.3 **
Sequence of differentiable functions and series uniform and pointwise
Let (f_n) be a sequence of differentiable functions f_n :[a,b] to R, such that the series
Σ from n=1 to infinity of f_n converges POINTWISE to a function f, and the series
Σ from n=1 to infinity of f’_n is UNIFORMLY summable.
Then f is differentiable and
f’(t) = Σ from n=1 to infinity of f’_n(t)
For all t in [a,b].
Theorem 8.5.4 WEIERSTRASS M-TEST
Let f_n from [a,b] to R be a sequence of functions.
Suppose we have a SUMMABLE sequence of real numbers (M_n) such that
| f_n (t) | is less than or equal to M_n for all n and all t in [a,b].
Then the sequence (f_n) is UNIFORMLY summmable.
Further for each t in [a,b] the series
Σ from n=1 to infinity of f_n(t)
Converges ABSOLUTELY.