Download OU B.Sc Computer Science 3rd Sem Real Analysis Important Questions

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Subject Title: Real Analysis Prepared by: Zikra Tarannum

Year: II Semester: III Updated on: 31-12-2020

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Unit - I: SEQUENCE AND SERIES

1. State and prove squeeze lemma.
2. State and prove Comparison test, Root test, Ratio test.
3. Convergent sequences are bounded.
4. a. If (sn) converges to s and (tn) converges to t, then (sn+tn) converges to s+t
   

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   b. If (sn) converges to s and (tn) converges to t, then (sn.tn) converges to st
   c. If (sn) converges to s ,if sn?0 for all n and if s?0 then (1/sn) converges to 1/s
   d. If (sn) converges to s and (tn) converges to t, if sn?0 for all n and if s?0 then (tn/sn) converges to t/s
5. A sequence is a convergent sequence iff it is a Cauchy sequence. (or) Every Cauchy sequence of real numbers is convergent
6. Convergent sequences are Cauchy Sequences.

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7. All bounded monotone sequences converge.
8. Every sequence (sn) has a monotone subsequence.
9. State and prove Bolzano - Weierstrass theorem (or) Every bounded sequence has a convergent subsequence
10. A series converges iff it satisfies the Cauchy criterion.
11. Problems on comparison, root, ratio and alternating series theorem.

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12. If sequence (sn) converges to a positive real number s and (tn) is any sequence, then lim sup(sntn)=s lim sup tn.
13. If the sequence (sn) converges, then every subsequence converges to the same limit.
14. Prove that the following
    a. lim (n->8) n^-p = 0 for p>0
    b. lim (n->8) aⁿ=0 if |a|<1
    

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    c. lim (n->8) n^1/n=1
    d. lim (n->8) a^1/n=1 for a>0
15. Let t₁ = 1 and t_n+1 = v(2+t_n) for n > 1, then assume (t_n) converges and find the limit.
16. Cauchy sequences are bounded.(or) Every Cauchy sequence of real numbers is bounded.
17. Every convergent sequence is bounded. Is converse true? Give example

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18. Prove that S 1/n^p convergent if p>1 by using integral test.

Unit - II: CONTINUITY

19. Define continuous function, uniformly continuous function.
20. Let f be a continuous real valued function on a closed interval [a, b]. Then f is a bounded function.
21. State and prove intermediate value theorem.

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22. If f is continuous on a closed interval [a, b], then f is uniformly continuous on [a, b].
23. If f is uniformly continuous on a set S and (s_n) is a Cauchy sequence in S, Then (f (s_n)) is a Cauchy sequence.
24. If f is uniformly continuous on a bounded set S, then f is a bounded function on S.
25. Problems on uniform continuity.
26. Let f( x ) = x² sin(1/x) for x?0 and f( 0 ) = 0. Prove f is continuous at 0.

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27. Suppose f is a real valued continuous function on R and f( a )f( b ) < 0 for Some a,b in R. Prove there exists x between a and b such that f( x )= 0.
28. Let f be a real valued function with dom (f) ? R. Then f is continuous at x₀ if and only if for every monotone sequence (x_n ) in dom( f ) converging to x₀, we have lim f(x_n)=f(x₀)
29. Prove x = cos x for some x in (0 ,p/2).
30. If f is uniformly continuous on its domain[a ,b] then show that f is continuous on Its domain [a,b].

Unit - III: DIFFERENTIATION

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31. State and prove Rolle’s theorem, Mean value theorem and Taylor’s theorem.
32. Suppose that f is differentiable at a then prove
    a. lim (x->a) [f(x) - f(a)]/(x-a) = f'(a)
    b. lim (h->0) [f(a+h) - f(a)]/h = f'(a)
33. Prove that | cosx-cosy|=|x-y|

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34. Prove that if 'f’ is differentiable at ‘a’, then ‘f’ is continuous at ‘a’
35. a. Find the Taylor series for cosx for all x
    b. Find the Taylor series for sinx for all x
36. Define derivative of a function ‘f’ at a point ‘a’
37. Find the following limits if they exists
    

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    a. lim (x->0) (x-sinx)/(1-cosx)
    b. lim (x->0+) x*ln(x)
    c. lim (x->0) (1-cos2x-2x²)/x²
    d. lim (x->8) x² e^-x
38. Use the definition of derivative to calculate the derivatives of the following functions at the indicated points
    

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    a. x³ at x=2
    b. g(x)=x+2 at x=a
    c. f (x)=x² cosx at x =0
    d. r(x)=(3x+4)/(2x-1) at x=1
39. Prove that lim (x->0) (a^x-1)/x = ln(a)

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40. Prove that lim (x->1) log_ax = 0
41. a. Show that x b. Show that f(x) = tan(x) is a strictly increasing function on (0,p/2)
    c. Show that x < (p/2) sinx for x?[0,p/2]
42. Find the Taylor series for sin hx=(e^x-e^-x)/2 and cos hx=(e^x+e^-x)/2
43. Prove that lim (n->8) (1 + (1/n))ⁿ = e .

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44. Prove that if f and g are differentiable on R, if f(0)=g(0) and if f'(x)
45. Find the following limits if they exists
    d.lim (x->0) (cosx)^1/x2

Unit-4 INTEGRATION

46. Define lower darboux sum, upper darboux sum, lower darboux integral, upper darboux integral and darboux integral: [ Hint:Darboux is also known as Riemann]

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47. Let f be a bounded function on [a , b]. if P and Q are partitions of [a , b] , Then L(f, P)= U(f, Q).
48. Let f be a bounded function on [a, b], then L (f)= U (f).
49. A bounded function f on [a , b] is integrable iff for each e>0 there exists a Partition P of [a , b] such that U(f, P) - L(f, P) < e.
50. Let f be a bounded function on [a, b]. if P and Q are partitions of [a, b] and P ? Q, then L(f, P)= L(f, Q)= U(f, Q)= U(f, P).
51. State and prove Cauchy criterion for integrability.

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52. A bounded function f on [a , b]is Riemann integrable iff it is a darboux Integrable.
53. a. Every monotone function f on [a , b] is integrable.
    b. Every continuous function f on [a , b] is integrable.
    c. Every constant function is integrable.
54. State and prove fundamental theorem of calculus.

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55. State and prove intermediate value theorem for integrals.
56. If f and g are integrable on [a , b], then prove that ?_a^b f = ?_a^b g , for all x in [a, b].
57. If f is integrable on [a , b], then |f| is integrable on [a , b] and |?_a^b f|=?_a^b |f| .
58. Show |?₇⁹ x⁴sin⁸(e^x)dx|= 2187/5
59. Let f be a function defined on [a, b]. if a < c < b and f is integrable on [a, c] and on [c, b], then prove that
    

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    (a). f is integrable on [a , b] and
    (b). ?_a^b f = ?_a^c f + ?_c^b f
60. Let f(x) = {x for rational/0 for irrational ,then calculate the upper and lower darboux integrals for f on the intervals [a , b]

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