Sequences of Functions, math homework help

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Sequences of Functions

Def. Let (fn) be a sequence of functions defined on a subset S of R. Then (fn) converges uniformly on S to a function f defined on S if

For each e > 0 there exists a number N such that for all x in S, for all n > N , | fn (x) -f(x) | < e.

#4.Let for x Î Let f (x) = 0.

Complete the following discussion and proof that (fn) converges uniformly to f on .

Discussion:

Suppose e is any positive real number.

We want to find N such that for all x Î and n > N, we have |fn(x) -f (x)| =< e

Note that since x ³ we haveand £ ____.( ____are numbers in simplest form)

______for all x Î

(___ is an expression involving an appropriate constant and the variable n only, no x)

So, we want______< e, which implies that n > _____.

Proof:

Let > 0. Choose N = _____. For all x Î , and n > N, we have

______ <_______= e, as desired.

(______ should be the expression involving e, before being simplified to get exactly e.)

#5.Let for x Î R.Let f(x) = 3x2.

Clearly, (fn) converges pointwise to f. But does it converge uniformly to f?

Fill in the blanks to carefully show that (fn) does not converge uniformly to f onR.

We must show: (the negation of the definition)

For _____ (all/some) e > 0, for _____ (all/some) N, for_____ (all/some) x in Rand _____ (all/some) n > N ,

| fn (x) - f(x) | __(<,>,£, ³)e.

Let = 1. Given any N , let n be a positive integer greater than N, and setx = en.

Then we have | fn (x) - f(x) | =______________________________________ (<,>,£, ³)1 = e.

(NOTE: In the _____________________substitute forfn (x) and f(x) and simplify, applying x = en.)


#6.Let for x Î [0, 1].

#6(a) State f (x) = lim fn(x).

#6 (b) Determine whether (fn) converges uniformly to f on[0, 1].Carefully justify your answer, either showing that the definition (shown on the previous page) holds, or proving that it cannot hold.

#7.Letfor x Î [-0.8, 1].

#7 (a) State a formula for f (x) = lim fn(x).(no explanation required)

#7 (b)(fn) does not converge uniformly to f on [-0.8,1].How can you deduce this fact without working hard? Please explain. HINT: Apply an appropriate theorem and an observation about the function f(x) you found in part (a).


Series of Functions (#8, 12 pts)

#8. Determine whether or not the given series of functions converges uniformly on the indicated set. Justify your answers with work/explanations . (HINT: Consider the Weierstrass M-Test and find an appropriate sequence Mn )

#8 (a)for x in R.

#8(b)for x Î.

#9.Show that it is possible to have a sequence of functions that are discontinuous at every point yet converge uniformly to a function that is continuous everywhere.

That is, state an example of a sequenceof functions (fn)and a function f satisfying all of the following:

  • Each fn is discontinuous at every real number.
  • (fn)converges uniformly to f .
  • f is continuous at every real number.

(explanation not required)

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Sequences of Functions Def. Let (fn) be a sequence of functions defined on a subset S of R. Then (fn) converges uniformly on S to a function f defined on S if For each  > 0 there exists a number N such that for all x in S, for all n > N , | fn (x)  f(x) | < . #4. Let 𝑓𝑛 (𝑥) = cos √𝑛𝑥 2 + √𝑛𝑥 1 for x [ , ∞). Let f (x) = 0. 9 1 Complete the following discussion and proof that (fn) converges uniformly to f on [9 , ∞). Discussion: Suppose  is any positive real number. 1 9 We want to find N such that for all x [ , ∞) and n > N, we have |fn(x)  f (x)| = Note that since x  | cos √𝑛𝑥 2 + √𝑛𝑥 | 1 9 we have 1 2 + √𝑛𝑥  √𝑥 ≥ ______ 1 √𝑛𝑥 = 1 ∙ 1 √𝑥 √𝑛 and  1 √𝑥  ____. | cos √𝑛𝑥 2 + √𝑛𝑥 − 0| <  ( ____ are numbers in simplest form) 1 9 ______ for all x  [ , ∞) (___ is an expression involving an appropriate constant and the variable n only, no x) So, we want ______ < , which implies that n > _____. Proof: 1 9 Let  > 0. Choose N = _____. For all x  [ , ∞), and n > N, we have |𝑓𝑛 (𝑥) − 𝑓(𝑥)| = | cos √𝑛𝑥 2 + √𝑛𝑥 − 0|  1 2 + √𝑛𝑥  1 √𝑛𝑥 = 1 ∙ 1 √𝑥 √𝑛  ______ < _______ = , as desired. (______ should be the expression involving , before being simplified to get exactly .) #5. Let 𝑓𝑛 (𝑥) = 2 ln(𝑥)+ 3𝑛𝑥 2 𝑛 = 2 ln(𝑥) 𝑛 + 3𝑥 2 for x  R. Let f(x) = 3x2. Clearly, (fn) converges pointwise to f. But does it converge uniformly to f? Fill in the blanks to carefully show that (fn) does not converge uniformly to f on R. We must show: (the negation of the definition) For _____ (all/some)  > 0, for _____ (all/some) N, for _____ (all/some) x in R and _____ (all/some) n > N , | fn (x)  f(x) | __ (,, ) . Let  = 1. Given any N , let n be a positive integer greater than N, and set x = en. Then we have | fn (x)  f(x) | = ____________________________________ __ (,, ) 1 = . (NOTE: In the _____________________ substitute for fn (x) and f(x) and simplify, applying x = en.) Page 1 of 3 #6. Let 𝑓𝑛 (𝑥) = 𝑥𝑛 4 √𝑛 for x  [0, 1]. #6(a) State f (x) = lim fn(x). #6 (b) Determine whether (fn) converges uniformly to f on [0, 1]. Carefully justify your answer, either showing that the definition (shown on the previous page) holds, or proving that it cannot hold. #7. Let 𝑓𝑛 (𝑥) = 𝑥𝑛 5 + 𝑥𝑛 for x  [0.8, 1]. #7 (a) State a formula for f (x) = lim fn(x). (no explanation required) #7 (b) (fn) does not converge uniformly to f on [0.8, 1]. How can you deduce this fact without working hard? Please explain. HINT: Apply an appropriate theorem and an observation about the function f(x) you found in part (a). Page 2 of 3 Series of Functions (#8, 12 pts) #8. Determine whether or not the given series of functions converges uniformly on the indicated set. Justify your answers with work/explanations . (HINT: Consider the Weierstrass M-Test and find an appropriate sequence Mn ) #8 (a) ∑ #8(b) ∑ sin(𝑒 𝑛𝑥 ) 𝑛4/3 1 𝑛𝑥 for x in R. for x [√3, ∞). #9. Show that it is possible to have a sequence of functions that are discontinuous at every point yet converge uniformly to a function that is continuous everywhere. That is, state an example of a sequence of functions (fn) and a function f satisfying all of the following: • • • Each fn is discontinuous at every real number. (fn) converges uniformly to f . f is continuous at every real number. (explanation not required) Page 3 of 3
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Attached.

Sequences of Functions
Def. Let (fn) be a sequence of functions defined on a subset S of R. Then (fn) converges uniformly on S to
a function f defined on S if
For each  > 0 there exists a number N such that for all x in S, for all n > N , | fn (x)  f(x) | < .
cos √𝑛𝑥
√𝑛𝑥

#4. Let 𝑓𝑛 (𝑥) = 2 +

1

for x [9 , ∞). Let f (x) = 0.
1

Complete the following discussion and proof that (fn) converges uniformly to f on [9 , ∞).
Discussion:
Suppose  is any positive real number.
1

cos √𝑛𝑥
√𝑛𝑥

We want to find N such that for all x [9 , ∞) and n > N, we have |fn(x)  f (x)| = |2 +
Note that since x 

cos √𝑛𝑥
|
√𝑛𝑥

|2 +



1
2 + √𝑛𝑥

1
9

x

we have



1
√𝑛𝑥

=

1
and
3

1
 3.
x

− 0| < 

( ____ are numbers in simplest form)

3
1
for all x  [9 , ∞)
n

1
1
∙ 𝑛
𝑥



(___ is an expression involving an appropriate
constant and the variable n only, no x)
So, we want

3
9
< , which implies that n > 2 .

n

Proof:

9
9
9
, where  2  represents the smallest integer greater than 2 . For all x 
2

 
 

Let  > 0. Choose N = 
1

[9 , ∞), and n > N, we have
|𝑓�...


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