Questions on Uniform Convergence with Answers

Practice Questions on Uniform Convergence with Answers

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Questions on Uniform Convergence with Answers


Q1. For n ≥1. Let gn(x) = sin2(x+1/n) , x∈[0,∞) and fn(x) = 0x gn(t) dt. Then

(a) {fn}converges point-wise to a function f on [0,∞), but does not converge uniformly on [0,∞)
(b) {fn} does not converge to any function on [0,∞)
(c) {fn} converges uniformly on [0,1)
(d) {fn} converges uniformly on [0,∞)


Q2. Let fn : [1,2] → [0,1] be given by fn(x) = (2-x)n for all non-negative integers n . Let f(x) = limn→∞ fn(x) for 1 ≤ x ≤ 2. Then which of the following is true ?

(a) f is a continuous function on [1,2]
(b) fn converges uniformly to f on [1,2] as n→∞
(c) limn→∞ 1∫2 fn(x)dx = 12 f(x) dx
(d) for any a∈(1,2) we have limn→∞ f’n(a) ≠ f'(a)


Q3. Let fn(x) = xn/(1+x) and gn(x) = xn/(1+nx) for x∈N. Then on the interval [0,1]

(a) both {fn} and {gn} converges uniformly
(b) neither {fn} nor {gn} converges uniformly
(c) {fn} converges uniformly but {gn} does not converge uniformly
(d) {gn} converges uniformly but {fn} does not converges uniformly


Q4. Which of the following sequence {fn}n=1 of functions does NOT converge uniformly on [0,1] ?

(a) fn(x) = (e-x)/n
(b) fn(x) = (1-x)n
(c) fn(x) = (x2 + nx)/n
(d) fn(x) = sin(nx+n)/n


Q5. Let fn(x) = x1/n for n∈[0,1]. Then

(a) limn→∞ fn(x) exists for all x∈[0,1]
(b) limn→∞ fn(x) defines a continuous function on [0,1]
(c) {fn} converges uniformly on [0,1]
(d) limn→∞ fn(x) = 0 for all x∈[0,1]


Q6. Let f(x) = Σn=1 [sin(nx)]/n2. Then

(a) limn→∞ f(x) = 0
(b) limn→∞ f(x) = 1
(c) limn→∞ f(x) = π2/6
(d) limn→∞ f(x) does not exist


Q7. Let {fn} be a sequence of real valued differentiable function on [a,b] such that fn(x) → f(x) as n→∞ for every x∈[a,b] and for some Riemann-integrable function f:[a,b] → R . Consider the statements :

P1 : {fn} converges uniformly
P2 : {f’n} converges unformly
P3 : ab fn(x) dx → ab f(x) dx
P4 : f is differentiable

Then which one of the following need NOT be true

(a) P1 implies P3
(b) P2 implies P1
(c) P2 implies P4
(d) P3 implies P1


Q8. Let fn(x)=(-x)n , x∈[0,1]. Then decide which of the following are true.

(a) there exist a point-wise convergent Subsequence of fn
(b) fn has non point-wise convergent subsequence.
(c) fn converges point-wise everywhere.
(d) fn has exactly one point-wise convergent subsequence.


Q9. Consider all sequences {fn} of real valued continuous functions on [0,∞). Identify which of the following statements are correct.

(a) If {fn} converges to f point-wise on [0,∞) then limn→∞ 0 fn(x) dx = 0 f(x) dx
(b) If {fn} converges to f uniformly on [0,∞) then limn→∞ 0 fn(x) dx = 0 f(x) dx
(c) If {fn} converges to f point-wise on [0,∞) then f is continuous on [0,∞)
(d) There exists a sequence of continuous functions {fn} on [0,∞) such that fn converges to f uniformly on [0,∞) but limn→∞ 0 fn(x) dx ≠ 0 f(x) dx


Q10. Which one of the following statements is true for the sequence of functions

fn(x) = 1/(n2+x2), n=1,2,3…. , x ∈[1/2,1]

(a) The sequence is monotonic and has 0 as the limit for all x ∈[1/2,1] as n→∞
(b) The sequence is not monotonic but has f(x) = 1/x2 as the limit as n→∞
(c) The sequençe is monotonic and has f(x) = 1/x2 as the limit as n→∞
(d) The sequence is not monotonic but has 0 as the limit.


Q11. Let fx(x) = sinx/√n, n = 1,2,3,… and x ∈ [-1,1]. Choose incorrect

(a) {fn(x)}n=1 does not converge uniformly in [-1,1]
(b) limn→∞ -11 fn(n) dx ≠ 0
(c) {fn(x)} does not converge uniformly in [-1,1]
(d) fn(x) , n = 1,2 is not uniformly continuous in [-1,1]


Q12. Consider two sequence {fn} and {gn} of functions where fn:[0,1] → R and gn: R → R are defined by fn(x) = xn and gn(x) = cos(n-x)π/2 if x∈[n-1, n+1] and gn(x) = 0 otherwise. Then

(a) Neither {fn} nor {gn} is uniformly convergent
(b) {fn} is not uniformly convergent but {gn} is
(c) {gn} is not uniformly convergent but {fn}
(d) Both {fn} and {gn} are uniformly convergent


Q13. Let {fn} be a sequence of continuous functions on R.

(a) If {fn} converges to f point-wise on R then limn→∞ -∞ fn(x) dx = -∞ f(x) dx
(b) If {fn} converges to f uniformly on R limn→∞ -∞ fn(x) dx = -∞ f(x) dx
(c) If {fn} converges to f uniformly on R then f is continuous on R
(d) There exists a sequence of continuous functions {fn} on R, such that {fn} converges to f uniformly on R , but limn→∞ -∞ fn(x) dx ≠ -∞ f(x) dx


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Answer Key

01.(c), (d)06.(a)11.(a), (b), (c), (d)
02.(c)07.(d)12.(a)
03.(d)08.(a), (b)13.(c), (d)
04.(b)09.(c), (d)
05.(a)10.(a)

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