Sequences An Series Of Functions

Question 1

When does a sequence of functions converge pointwise

Answer 1

A sequence of functions (fk)k converge pointwise to f on A as lim k tends to inf if sequence (fk(x)) converges to f(x) for every x in A.
If for every x in A and every epsilon > 0 exists N in N s.t for every k >= N we have mod(fk(x) - f(x)) < epsilon
There is a fixed x, k goes to inf

Question 2

When does a sequence converges uniformly

Answer 2

A sequence converges uniformly to f on A if:
For all epsilon > 0 exists N in N s.t for all k >= N for all x in A mod(fk(x)-f(x)) < epsilon
Different because epsilon can’t depend on x
X not fixed, K fixed at first

Question 3

When does fk converge to f uniformly

Answer 3

Fk converges to f uniformly on A when:
Lim k tends to inf (sup mod(fk(x) - f(x))) = 0
Supremum because can choose x = epsilon

Question 4

What is the sequence of partial sums

Answer 4

Sequence of partial sums is:
Sn = sum k=1 to n fk

Question 5

When does a series sum k=1 to inf fk converge pointwise

Answer 5

A series sum k=1to inf converges pointwise when the sequence (Sn)n converges to f pointwise on A

Question 6

When does series sum k=1 to inf fk converge uniformly

Answer 6

Series sum k=1 to inf fk converges uniformly when series (Sn)n converge to f uniformly on A

Question 7

What does uniform convergence imply

Answer 7

Uniform convergence implies pointwise convergence, but vice versa isn’t necessarily true

Question 8

What is Cauchy criterion for uniform convergence

Answer 8

Cauchy criterion fo uniform convergence is:
A sequence of functions (fk) fk : A to R converges uniformly on A iff for every epsilon > 0 exists an integer N s.t if j,m >= N and x is in A then mod(fj(x) -fm(x)) < epsilon.
Useful because don’t need to find limit to prove

Question 9

What is the weierstrass M-test

Answer 9

Weierstrass M-test is:
Sup(Mod(fk(x))) <= Mk for every x, sum k=1 to inf Mk < inf then series fk converges uniformly on A

Question 10

If uniform convergence of a series is proved by weierstrass then what else is proved

Answer 10

If uniform convergence is proved by weierstrass M-test then series also converges absolutely, but uniform convergence doesn’t moly absolute convergence

Question 11

When is weierstrass M-test not applicable

Answer 11

Weierstrass M-test is not applicable when there is a series that converges uniformly but not absolutely

Question 12

If fk tends to f uniformly as k tends to inf, then what is f

Answer 12

If fk tends uniformly to f as k tends to inf, then f is continuous on interval