Download PTU B.Sc CS-IT 2020 March 3rd Sem 71774 Sequence Series And Calculus Question Paper

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Roll No. [T Total No. of Pages : 02

Total No. of Questions : 07

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B.Sc. (Computer Science) (2013 & Onwards) (Sem.-3)

SEQUENCE SERIES AND CALCULUS

Subject Code : BCS-302

M.Code : 71774

Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains SIX questions carrying TEN marks each and students have to attempt ANY FOUR questions.

SECTION-A

1. Write briefly :
   1. If a sequence is divergent to 8, then it is bounded below but not bounded above.

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   2. Prove that the sequence <1/n> is convergent.
   3. Show that the series : 1 - 1/2! + 1/3! + ... is Convergent.
   4. State Raabe’s test.
   5. Prove that the series Su_n is divergent where u_n = n/(n+1)
   6. State the first mean value theorem of integral calculus.

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   7. State comparison test in limit form for convergence of improper integral ?_a^b f(x)dx.
   8. Show that ?₀⁸ x*e^-x dx= 8
   9. Express ?₀¹ x² (1-x²)ⁿ dx as a beta function.
   10. Compute ?₀¹ f(x)dx where f(x) =|x|.

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SECTION-B

2. a) Every cauchy sequence of real numbers is convergent.
3. b) if S a_n =1, where |r| <1, then lim a_n=0
4. a) Test the convergence of the series S (n/(n+1)) * xⁿ.

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5. b) if Su_n is convergent, show that S (u_n/(1-u_n)) is also convergent. (u_n >0,u_n ?1)
6. a) Show that the series : 1 - 1/2 + 1/3 - 1/4 + ... is conditionally convergent.
7. b) State and prove cauchy’s general principle of convergence.
8. a) Prove that if a function is monotonic on [a, b], then show then it is Riemann integrable on [a, b].
9. b) If 0 <x < 1, then show that 1/(1-x) > log (1-x) > x.

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10. a) Check for convergence the improper integral ?₀¹ x^m-1 (1-x)^n-1 dx where m, n are real numbers.
11. b) State and prove cauchy’s test for convergence of ?_a^b f(x)dx at a.
12. a) Show that : B(m,n)=?₀⁸ x^m-1/(1+x)^(m+n)dx; m>0,n>O.
13. b) Show that G(1/2) = vp.

NOTE : Disclosure of identity by writing mobile number or making passing request on any page of Answer sheet will lead to UMC case against the Student.

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