Advanced Calculus (I)

Advanced Calculus (I)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

WEN-CHINGLIEN Advanced Calculus (I)

7.2 Uniform convergence of series

Definition

Let fk be a sequence of real functions defined on some set E and set

sn(x ) :=

n

X

k =1

fk(x ), x ∈ E , n ∈N.

(i)

The series

∞

P

k =1

fk is said to converges pointwise on E if and only if the sequence sn(x ) converges pointwise on E as n → ∞.

WEN-CHINGLIEN Advanced Calculus (I)

7.2 Uniform convergence of series

Definition

Let fk be a sequence of real functions defined on some set E and set

sn(x ) :=

n

X

k =1

fk(x ), x ∈ E , n ∈N.

(i)

The series

∞

P

k =1

fk is said to converges pointwise on E if and only if the sequence sn(x ) converges pointwise on E as n → ∞.

WEN-CHINGLIEN Advanced Calculus (I)

7.2 Uniform convergence of series

Definition

Let fk be a sequence of real functions defined on some set E and set

sn(x ) :=

n

X

k =1

fk(x ), x ∈ E , n ∈N.

(i)

The series

∞

P

k =1

fk is said to converges pointwise on E if and only if the sequence sn(x ) converges pointwise on E as n → ∞.

WEN-CHINGLIEN Advanced Calculus (I)

7.2 Uniform convergence of series

Definition

Let fk be a sequence of real functions defined on some set E and set

sn(x ) :=

n

X

k =1

fk(x ), x ∈ E , n ∈N.

(i)

The series

∞

P

k =1

fk is said to converges pointwise on E if and only if the sequence sn(x ) converges pointwise on E as n → ∞.

WEN-CHINGLIEN Advanced Calculus (I)

(ii)

The series

∞

P

k =1

fk is said to converge uniformly on E if and only if the sequence sn(x ) converges uniformly on E as n → ∞.

(iii)

The series

∞

P

k =1

fk is said to converge absolutely (pointwise) on E if and only if

∞

P

k =1

|f_k(x )| converges for each x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)

The series

∞

P

k =1

fk is said to converge uniformly on E if and only if the sequence sn(x ) converges uniformly on E as n → ∞.

(iii)

The series

∞

P

k =1

fk is said to converge absolutely (pointwise) on E if and only if

∞

P

k =1

|f_k(x )| converges for each x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)

The series

∞

P

k =1

fk is said to converge uniformly on E if and only if the sequence sn(x ) converges uniformly on E as n → ∞.

(iii)

The series

∞

P

k =1

fk is said to converge absolutely (pointwise) on E if and only if

∞

P

k =1

|f_k(x )| converges for each x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)

The series

∞

P

k =1

fk is said to converge uniformly on E if and only if the sequence sn(x ) converges uniformly on E as n → ∞.

(iii)

The series

∞

P

k =1

fk is said to converge absolutely (pointwise) on E if and only if

∞

P

k =1

|f_k(x )| converges for each x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)

The series

∞

P

k =1

fk is said to converge uniformly on E if and only if the sequence sn(x ) converges uniformly on E as n → ∞.

(iii)

The series

∞

P

k =1

fk is said to converge absolutely (pointwise) on E if and only if

∞

P

k =1

|f_k(x )| converges for each x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and let {fk} be a sequence of real functions defined on E.

(i)

Suppose that x0 ∈ E and that each fk is continuous at x0 ∈ E. If f =

∞

P

k =1

fk converges uniformly on E, then f is continuous at x0∈ E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and let {fk} be a sequence of real functions defined on E.

(i)

Suppose that x0 ∈ E and that each fk is continuous at x0 ∈ E. If f =

∞

P

k =1

fk converges uniformly on E, then f is continuous at x0∈ E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and let {fk} be a sequence of real functions defined on E.

(i)

Suppose that x0 ∈ E and that each fk is continuous at x0 ∈ E. If f =

∞

P

k =1

fk converges uniformly on E, then f is continuous at x0∈ E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and let {fk} be a sequence of real functions defined on E.

(i)

Suppose that x0 ∈ E and that each fk is continuous at x0 ∈ E. If f =

∞

P

k =1

fk converges uniformly on E, then f is continuous at x0∈ E.

WEN-CHINGLIEN Advanced Calculus (I)

(ii)[Term-by-term integration]

Suppose that E = [a, b] and that each fk is integrable on [a,b]. If f =

∞

P

k =1

fk converges uniformly on [a,b], then f is integrable on [a,b] and

Z b a

∞

X

k =1

fk(x )dx =

∞

X

k =1

Z b a

fk(x )dx .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)[Term-by-term integration]

Suppose that E = [a, b] and that each fk is integrable on [a,b]. If f =

∞

P

k =1

fk converges uniformly on [a,b], then f is integrable on [a,b] and

Z b a

∞

X

k =1

fk(x )dx =

∞

X

k =1

Z b a

fk(x )dx .

WEN-CHINGLIEN Advanced Calculus (I)

(ii)[Term-by-term integration]

Suppose that E = [a, b] and that each fk is integrable on [a,b]. If f =

∞

P

k =1

fk converges uniformly on [a,b], then f is integrable on [a,b] and

Z b a

∞

X

k =1

fk(x )dx =

∞

X

k =1

Z b a

fk(x )dx .

WEN-CHINGLIEN Advanced Calculus (I)

(iii)[Term-by-term differentation]

Suppose that E is bounded, open interval and that each fk

is differentiable on E. If

∞

P

k =1

fk converges at some x0 ∈ E, and

∞

P

k =1

f_k⁰ converges uniformly on E, then f :=

∞

P

k =1

fk

converges uniformly on E, f is differentiable on E, and

∞

X

k =1

fk(x )

!0

=

∞

X

k =1

f_k⁰(x )

for x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(iii)[Term-by-term differentation]

Suppose that E is bounded, open interval and that each fk

is differentiable on E. If

∞

P

k =1

fk converges at some x0 ∈ E, and

∞

P

k =1

f_k⁰ converges uniformly on E, then f :=

∞

P

k =1

fk

converges uniformly on E, f is differentiable on E, and

∞

X

k =1

fk(x )

!0

=

∞

X

k =1

f_k⁰(x )

for x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

(iii)[Term-by-term differentation]

Suppose that E is bounded, open interval and that each fk

is differentiable on E. If

∞

P

k =1

fk converges at some x0 ∈ E, and

∞

P

k =1

f_k⁰ converges uniformly on E, then f :=

∞

P

k =1

fk

converges uniformly on E, f is differentiable on E, and

∞

X

k =1

fk(x )

!0

=

∞

X

k =1

f_k⁰(x )

for x ∈ E .

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR, let fk :E → R, k ∈N, and let Mk ≥ 0 satisfy

∞

P

k =1

Mk < ∞. If |fk(x )| ≤ Mk for k ∈N and x ∈ E , then

∞

P

k =1

fk converges absolutely and uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR, let fk :E → R, k ∈N, and let Mk ≥ 0 satisfy

∞

P

k =1

Mk < ∞. If |fk(x )| ≤ Mk for k ∈N and x ∈ E , then

∞

P

k =1

fk converges absolutely and uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and suppose that fk,gk :E → R, k ∈N. If

n

X

k =1

fk(x )     

≤ M < ∞

for n ∈N and x ∈ E , and if g_k ↓ 0 uniformly on E as k → ∞, then

∞

P

k =1

fkgk converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

Let E be a nonempty subset ofR and suppose that fk,gk :E → R, k ∈N. If

n

X

k =1

fk(x )     

≤ M < ∞

for n ∈N and x ∈ E , and if g_k ↓ 0 uniformly on E as k → ∞, then

∞

P

k =1

fkgk converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

A double series is convergent if and only if

∞

X

j=1

ak_j converges for each k ∈N

and

∞

X

k =1

∞

X

j=1

akj := lim

N→∞

N

X

k =1





∞

X

j=1

akj



 exists and is finite.

WEN-CHINGLIEN Advanced Calculus (I)

A double series is convergent if and only if

∞

X

j=1

ak_j converges for each k ∈N

and

∞

X

k =1

∞

X

j=1

akj := lim

N→∞

N

X

k =1





∞

X

j=1

akj



 exists and is finite.

WEN-CHINGLIEN Advanced Calculus (I)

A double series is convergent if and only if

∞

X

j=1

ak_j converges for each k ∈N

and

∞

X

k =1

∞

X

j=1

akj := lim

N→∞

N

X

k =1





∞

X

j=1

akj



 exists and is finite.

WEN-CHINGLIEN Advanced Calculus (I)

Let akj ∈ R for k , j ∈ N and suppose that

Aj =

∞

X

k =1

|ak_j| < ∞

for each j ∈N. If

∞

P

j=1

converges (i.e., the double sum converges absolutely), then

∞

X

k =1

∞

X

j=1

akj =

∞

X

j=1

∞

X

k =1

akj.

WEN-CHINGLIEN Advanced Calculus (I)

Let akj ∈ R for k , j ∈ N and suppose that

Aj =

∞

X

k =1

|ak_j| < ∞

for each j ∈N. If

∞

P

j=1

converges (i.e., the double sum converges absolutely), then

∞

X

k =1

∞

X

j=1

akj =

∞

X

j=1

∞

X

k =1

akj.

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N,define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis, fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N, define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis, fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N,define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis,fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N, define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis, fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N, define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis,fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

Proof:

Let E = {0, 1,¹₂,¹₃, · · · }. For each j ∈N, define a function fj on E by

fj(0) =

∞

X

k =1

ak_j, fj

 1 n



=

∞

X

k =1

ak_j, n ∈ N.

By hypothesis, fj(0) exists and by the definition of series convergence,

n→∞lim fj

 1 n



=fj(0);

WEN-CHINGLIEN Advanced Calculus (I)

i.e., fj is continuous at 0 ∈ E for each j ∈

since |fj(x )| ≤ Aj for all x ∈ E and j ∈N, The Weierstrass M-Test implies that

f (x ) :=

∞

X

j=1

fj(x )

converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

i.e., fj is continuous at 0 ∈ E for each j ∈ Moreover, since |fj(x )| ≤ Aj for all x ∈ E and j ∈N,The Weierstrass M-Test implies that

f (x ) :=

∞

X

j=1

fj(x )

converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

i.e., fj is continuous at 0 ∈ E for each j ∈

since |fj(x )| ≤ Aj for all x ∈ E and j ∈N, The Weierstrass M-Test implies that

f (x ) :=

∞

X

j=1

fj(x )

converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

i.e., fj is continuous at 0 ∈ E for each j ∈

since |fj(x )| ≤ Aj for all x ∈ E and j ∈N,The Weierstrass M-Test implies that

f (x ) :=

∞

X

j=1

fj(x )

converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

i.e., fj is continuous at 0 ∈ E for each j ∈

since |fj(x )| ≤ Aj for all x ∈ E and j ∈N, The Weierstrass M-Test implies that

f (x ) :=

∞

X

j=1

fj(x )

converges uniformly on E.

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

from the sequential characteristization of continuity (Theorem 3.21) that f (1

n) →f (0) as n → ∞. Therefore,

∞

P

k =1

∞

P

j=1

ak_j = lim

n→∞

n

P

k =1

∞

P

j=1

ak_j

= lim

n→∞

∞

P

j=1 n

P

k =1

akj

= lim

n→∞

∞

P

j=1

fj

 1 n



=f (0) =

∞

P

j=1

∞

P

k =1

akj.2

WEN-CHINGLIEN Advanced Calculus (I)

Thank you.

WEN-CHINGLIEN Advanced Calculus (I)