Chapter 6

Question 1

Pointwise Convergence (definition A)-

For each n in N, let f_n be a function defined on a set A c R. The sequence (f_n) converges pointwise on A if…

Answer 1

for all x in A, the sequence of real numbers f_n(x) converges to f(x)

Question 2

for all x in A, the sequence of real numbers f_n(x) converges to f(x)

Answer 2

Pointwise Convergence (definition A)-

For each n in N, let f_n be a function defined on a set A c R. The sequence (f_n) converges pointwise on A if…

Question 3

Uniform Convergence-

Let (f_n) be a sequence of functions defined on a set A c R. Then, (f_n) converges uniformly on A to a limit function f defined on A if…

Answer 3

for every ε > 0, there exists an N in N such that

|f_n(x) - f(x)| < ε

whenever n > N and x in A.

Question 4

for every ε > 0, there exists an N in N such that

|f_n(x) - f(x)| < ε

whenever n > N and x in A.

Answer 4

Uniform Convergence-

Let (f_n) be a sequence of functions defined on a set A c R. Then, (f_n) converges uniformly on A to a limit function f defined on A if…

Question 5

Pointwise Convergence (definition B)-

Let f_n be a sequence of functions defined on a set A c R. Then, (f_n) converges pointwise on A to a limit f defined on A if…

Answer 5

for every ε > 0, and x in A, there exists an N in N (perhaps dependent on x) such that

|f_n(x) - f(x)|

whenever n > N.

Question 6

for every ε > 0, and x in A, there exists an N in N (perhaps dependent on x) such that

|f_n(x) - f(x)| < ε

whenever n > N.

Answer 6

Pointwise Convergence (definition B)-

Let f_n be a sequence of functions defined on a set A c R. Then, (f_n) converges pointwise on A to a limit f defined on A if…

Question 7

Cauchy Criterion for Uniform Convergence-

A sequence of function (f_n) defined on a set A c R converges uniformly if and only if…

Answer 7

for every ε > 0 there exists an N in N such that

|f_n(x) - f_m(x)| < ε

whenever m,n > N and x in A.

Question 8

for every ε > 0 there exists an N in N such that

|f_n(x) - f_m(x)| < ε

whenever m,n > N and x in A.

Answer 8

Cauchy Criterion for Uniform Convergence-

A sequence of function (f_n) defined on a set A c R converges uniformly if and only if…

Question 9

Continuous Limit Theorem-

Let (f_n) be a sequence of functions defined on A c R that converges uniformly on A to the function f. If each f_n is continuous at c in A…

Answer 9

then f is continuous at c.

Question 10

Differentiable Limit Theorem-

Let f_n converge to f pointwise on the closed interval [a,b] and assume that each f_n is differentiable. If (f_n’) converges uniformly on [a,b] to a function g, then…

Answer 10

the function f is differentiable and f’ = g.

Question 11

the function f is differentiable and f’ = g.

Answer 11

Differentiable Limit Theorem-

Let f_n converge to f pointwise on the closed interval [a,b] and assume that each f_n is differentiable. If (f_n’) converges uniformly on [a,b] to a function g, then…

Question 12

Let (f_n) be a sequence of differentiable functions defined on the closed interval [a,b] and assume (f_n’) converges uniformly on [a,b] to a function g. If there exists a point x₀ in [a,b] where f_n(x₀) is convergent, then…

Answer 12

(f_n) converges uniformly on [a,b]. Moreover, the limit function f = lim f_n is differentiable and satisfies f’ = g.

Question 13

(f_n) converges uniformly on [a,b]. Moreover, the limit function f = lim f_n is differentiable and satisfies f’ = g.

Answer 13

Let (f_n) be a sequence of differentiable functions defined on the closed interval [a,b] and assume (f_n’) converges uniformly on [a,b] to a function g. If there exists a point x₀ in [a,b] where f_n(x₀) is convergent, then…

Question 14

Pointwise Convergence for Series-

For each n in N, let f_n and f be functions defined on a set A c R. The infinite series Σf_n(x) converges pointwise on A to f(x) if…

Answer 14

the sequence s_k(x) of partial sums where

s_k(x) = f₁ + …. + f_k

converges pointwise to f(x). The series converges uniformly on A to f if the sequence s_k(x) converges uniformly on A to f(x).

Question 15

the sequence s_k(x) of partial sums where

s_k(x) = f₁ + …. + f_k

converges pointwise to f(x). The series converges uniformly on A to f if the sequence s_k(x) converges uniformly on A to f(x).

Answer 15

Pointwise Convergence for Series-

For each n in N, let f_n and f be functions defined on a set A c R. The infinite series Σf_n(x) converges pointwise on A to f(x) if…

Question 16

Term-by-Term Continuity Theorem-

Let f_n be continuous functions defined on a set A c R and assume Σf_n converges uniformly on A to a function f, then…

Answer 16

f is continuous on A.

Question 17

Term-by-Term Differentiability Theorem-

Let f_n be differentiable functions on an inteval A and assume Σf_n‘(x) converges uniformly to a limit g(x) on A. If there exists a point x₀ in [a,b] where Σf_n(x₀) converges, then…

Answer 17

the series Σf_n(x₀) converges uniformly to a differentiable to a differentiable function f(x) satisfying f’(x) = g(x) on A. In other words,

f(x) = Σf_n(x)

and

f’(x) = Σf_n‘(x)

Question 18

the series Σf_n(x₀) converges uniformly to a differentiable to a differentiable function f(x) satisfying f’(x) = g(x) on A. In other words,

f(x) = Σf_n(x)

and

f’(x) = Σf_n‘(x)

Answer 18

Term-by-Term Differentiability Theorem-

Let f_n be differentiable functions on an inteval A and assume Σf_n‘(x) converges uniformly to a limit g(x) on A. If there exists a point x₀ in [a,b] where Σf_n(x₀) converges, then…

Question 19

Cauchy Criterion for Uniform Convergence of Series-

A series Σf_n converges on A c R if and only if…

Answer 19

for every ε > 0 there exists an N in N such that

|f_m+1(x) + f_m+2(x) … f_n(x)|

wheneve n > m > N and x in A.

Question 20

for every ε > 0 there exists an N in N such that

|f_m+1(x) + f_m+2(x) … f_n(x)| < ε

whenever n > m > N and x in A.

Answer 20

Cauchy Criterion for Uniform Convergence of Series-

A series Σf_n converges on A c R if and only if…

Question 21

Weierstrass M-Type-

For each n in N, let f_n be a function defined on a set A c R and let M_n > 0 be a real number where

|f_n(x)| < M_n

for all x in A. If ΣM_n converges, then…

Answer 21

Σf_n converges uniformly on A.