Practice Questions on Uniform Convergence with Answers

**Questions on Uniform Convergence with Answers**

**Q1. For n ≥1. Let g _{n}(x) = sin^{2}(x+^{1}/_{n}) , x∈[0,∞) and f_{n}(x) = _{0}∫^{x} g_{n}(t) dt. Then**

(a) {f_{n}}converges point-wise to a function f on [0,∞), but does not converge uniformly on [0,∞)

(b) {f_{n}} does not converge to any function on [0,∞)

(c) {f_{n}} converges uniformly on [0,1)

(d) {f_{n}} converges uniformly on [0,∞)

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

(a) f is a continuous function on [1,2]

(b) f_{n} converges uniformly to f on [1,2] as n→∞

(c) lim_{n→∞} 1∫2 fn(x)dx = _{1}∫^{2} f(x) dx

(d) for any a∈(1,2) we have lim_{n→∞} f’_{n}(a) ≠ f'(a)

**Q3. Let f _{n}(x) = x^{n}/_{(1+x)} and g_{n}(x) = x^{n}/_{(1+nx)} for x∈N. Then on the interval [0,1]**

(a) both {f_{n}} and {g_{n}} converges uniformly

(b) neither {f_{n}} nor {g_{n}} converges uniformly

(c) {f_{n}} converges uniformly but {g_{n}} does not converge uniformly

(d) {g_{n}} converges uniformly but {f_{n}} does not converges uniformly

**Q4. Which of the following sequence {f _{n}}^{∞}_{n=1} of functions does NOT converge uniformly on [0,1] ?**

(a) f_{n}(x) = (e^{-x})/_{n}

(b) f_{n}(x) = (1-x)^{n}

(c) f_{n}(x) = (x^{2} + nx)/_{n}

(d) f_{n}(x) = sin(nx+n)/_{n}

**Q5. Let f _{n}(x) = x^{1/n} for n∈[0,1]. Then**

(a) lim_{n→∞} f_{n}(x) exists for all x∈[0,1]

(b) lim_{n→∞} f_{n}(x) defines a continuous function on [0,1]

(c) {f_{n}} converges uniformly on [0,1]

(d) lim_{n→∞} f_{n}(x) = 0 for all x∈[0,1]

**Q6. Let f(x) = Σ ^{∞}_{n=1} ^{[sin(nx)]}/n^{2}. Then**

(a) lim_{n→∞} f(x) = 0

(b) lim_{n→∞} f(x) = 1

(c) lim_{n→∞} f(x) = π^{2}/_{6}

(d) lim_{n→∞} f(x) does not exist

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

**P _{1} : {f_{n}} converges uniformly**

**P**

_{2}: {f’_{n}} converges unformly**P**

_{3}:_{a}∫^{b}fn(x) dx →_{a}∫^{b}f(x) dx**P**

_{4}: f is differentiable**Then which one of the following need NOT be true**

(a) P_{1} implies P_{3}

(b) P_{2} implies P_{1}

(c) P_{2} implies P_{4}

(d) P_{3} implies P_{1}

**Q8. Let f _{n}(x)=(-x)^{n} , x∈[0,1]. Then decide which of the following are true.**

(a) there exist a point-wise convergent Subsequence of f_{n}

(b) f_{n} has non point-wise convergent subsequence.

(c) f_{n} converges point-wise everywhere.

(d) f_{n} has exactly one point-wise convergent subsequence.

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

(a) If {f_{n}} converges to f point-wise on [0,∞) then lim_{n→∞ 0}∫^{∞} f_{n}(x) dx = _{0}∫^{∞} f(x) dx

(b) If {f_{n}} converges to f uniformly on [0,∞) then lim_{n→∞} _{0}∫^{∞} f_{n}(x) dx = _{0}∫^{∞} f(x) dx

(c) If {f_{n}} converges to f point-wise on [0,∞) then f is continuous on [0,∞)

(d) There exists a sequence of continuous functions {f_{n}} on [0,∞) such that f_{n} converges to f uniformly on [0,∞) but lim_{n→∞} _{0}∫^{∞} f_{n}(x) dx ≠ _{0}∫^{∞} f(x) dx

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

**f _{n}(x) = ^{1}/(n^{2}+x^{2}), 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}/x^{2} as the limit as n→∞

(c) The sequençe is monotonic and has f(x) = ^{1}/x^{2} as the limit as n→∞

(d) The sequence is not monotonic but has 0 as the limit.

**Q11. Let f _{x}(x) = ^{sinx}/_{√n}, n = 1,2,3,… and x ∈ [-1,1]. Choose incorrect**

(a) {f_{n}(x)}^{∞}_{n=1} does not converge uniformly in [-1,1]

(b) lim_{n→∞} _{-1}∫^{1} f_{n}(n) dx ≠ 0

(c) {f_{n}(x)} does not converge uniformly in [-1,1]

(d) f_{n}(x) , n = 1,2 is not uniformly continuous in [-1,1]

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

(a) Neither {f_{n}} nor {g_{n}} is uniformly convergent

(b) {f_{n}} is not uniformly convergent but {g_{n}} is

(c) {g_{n}} is not uniformly convergent but {f_{n}}

(d) Both {f_{n}} and {g_{n}} are uniformly convergent

**Q13. Let {f _{n}} be a sequence of continuous functions on R.**

(a) If {f_{n}} converges to f point-wise on R then lim_{n→∞} _{-∞}∫^{∞} f_{n}(x) dx = _{-∞}∫^{∞} f(x) dx

(b) If {f_{n}} converges to f uniformly on R lim_{n→∞} _{-∞}∫^{∞} f_{n}(x) dx = _{-∞}∫^{∞} f(x) dx

(c) If {f_{n}} converges to f uniformly on R then f is continuous on R

(d) There exists a sequence of continuous functions {f_{n}} on R, such that {f_{n}} converges to f uniformly on R , but lim_{n→∞} _{-∞}∫^{∞} f_{n}(x) dx ≠ _{-∞}∫^{∞} f(x) dx

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*Answer Key*

*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) |