B.Sc. First Year Convergence Previous Year Questions | Previous Year Questions For B.Sc. Part 1 | Question Bank For B.Sc. Mathematics

 B.Sc. Part 1 Mathematics 

 CONVERGENCE 

Ques1: Define a bounded sequence.  [Kanpur B.Sc. 2008]

Ques2: Give an example of a sequence which is not bounded.  [Kanpur B.Sc. 1993]

Ques3: Give an example of a monotonic increasing sequence which is : (i) bounded, and (ii) unbounded.  [Kanpur B.Sc. 2010]

Ques4: Define monotonic sequences by giving one example of each.  [Kanpur B.Sc. 1992; Garhwal 1991]

Ques5: Give an example in each of the following cases: 

(i)   Monotonic increasing sequence which is not bounded above.  [Kanpur B.Sc. 1992]

(ii)  A sequence which is bounded above as-well-as bounded below but not monotonic.  [Kanpur B.Sc. 1992, 94]

(iii) A sequence which is not monotonic.  [Kanpur B.Sc. 1992]

Ques6: Show that the sequence <n - 1 / n> is monotonic increasing.  [Kashi 2012]

Ques7: Show that < 1/n > is a bounded monotonic sequence.  [Avadh 2013]

Ques8: Show that a convergent sequence does not converge to two different limits.  [Kanpur B.Sc. 2003, 07]

Ques9: Prove that ''If a sequence is convergent, then its limit is unique''.  [Kanpur B.Sc. 1997, 2004, 05, 10; Avadh 2011, 14; Garhwal 1991; Purvanchal 2009, 14]

Ques10: Show that ''Every convergent sequence is bounded. But the converse need not be true''.  [Kanpur B.Sc. 1995, 98, 2006, 08; Kashi 2011; Purvanchal 1991; Gorakh. 1986]

Ques11: Show that ''A monotonic sequence is convergent if it is bounded''.  [Kanpur B.Sc. 2000]

Ques12: Prove : A monotonic sequence is convergent if and only if it is bounded.  [Kanpur B.Sc. 1997, 98, 2012; Avadh 2007; Purvanchal 2009; Kashi 2012]

Ques13: Show that : ''A monotonically increasing sequence in R is convergent if and only if it is bounded above.  [Purvanchal 2010]

Ques14: Show that the sequence converges to a limit independent of k and find the limit.  [Avadh 2008]

Ques15: Define convergent sequence by giving one example.  [Kanpur B.Sc. 1992; Garhwal 1991]

Ques16: Define limit of a sequence. Illustrate by giving an example.  [Kanpur B.Sc. 1994, 96, 98]

Ques17: Define the convergence of a sequence. Prove that a convergent sequence is bounded but the converse may not be true.  [Purvanchal 1991]

Ques18: Give an example of a sequence which is not convergent.  [Kanpur B.Sc. 1994]
Ques19: Give an example of a monotonic decreasing sequence which is not convergent.  [Kanpur B.Sc. 1993]
Ques20: Give an example of a monotonic increasing sequence which is convergent.  [Kanpur B.Sc. 1993]

Ques21: Using definition of the limit of a sequence show that the limit of the sequence < 2n / (n+3) > is 2.  [Meerut 2014]

Ques22: Prove that the converse of every Cauchy sequence is bounded.  [Kanpur B.Sc. 2013; Purvanchal 2008, 11]
Ques23: Prove that the converse of every Cauchy sequence in R is convergent.  [Kanpur B.Sc. 1995; Garhwal 1992, 96; Purvanchal B.Sc. 1990]
Ques24: Prove that a sequence of real number is convergent if and only if it is a Cauchy sequence.  [Kanpur B.Sc. 2003; Rohil. 2013; Purvanchal B.Sc. 1991]
Ques25: State and prove Cauchy's general principle of convergence.  [Kanpur B.Sc. 2013]
Ques26: Define Cauchy sequence of real numbers.  [Purvanchal 2014; Kanpur B.Sc. 1994(R)]
Ques27: Define Cauchy sequence. Prove that a sequence of real numbers is convergent if it is a Cauchy sequence.  [Garhwal 1997]
Ques28: State and prove Cauchy's general principle of convergence.  [Purvanchal 1991]
Ques29: What is alternating series.  [Meerut 2014]
Ques30: Define convergent series.  [Kanpur B.Sc. 1990, 96]
Ques31: Test the nature of the series :
2-2+2-2+2-2+... ... .  [Kanpur B.Sc. 2014]
Ans: Oscillatory.
Ques32: Show that the given series is divergent :
(a/b) + (a+x/b+x) + (a+2x/b+2x) + (a+3x/b+3b) + ... .  [Kanpur B.Sc. 1996]
Ques33: State and prove p-test for convergence and divergence of an infinite series.  [Kanpur B.Sc. 2005]
Ques34: Test the nature of the series :
1+1/2+1/3+1/4+... ... .  [Bundel. 2014]
Ans: Divergent.
Ques35: Test the convergence of the series
1-1/ ---.  [Kanpur B.Sc. 2007; Rohil. 1996]
Ques36: Test the convergence of the series :
(1/7-1/log7)-(1/7-1/log8)+(1/7-1/log9)-(1/7-1/log10)+(1/7-1/log11)- ... .  [Kanpur B.Sc. 1995]
Ques37: Define convergent, divergent and oscillatory series with examples.  [Kanpur B.Sc. 1990, 96, 2012; Agra 1998]
Ques38: Test the convergence of the following series :
(i)   1-1/2 + 1/3 - 1/4 + ... ... .  [Meerut 2014]
(ii)  1/1.2 - 1/3.4 + 1/5.6 - 1/7.8 + .... . [Purvanchal 2014; Bundel. 2008]
(iii) 1 - 1/2√2 + 1/3√3 -1/4√4 + ... .  [Kanpur B.Sc. 1995, 96, 98]
Ques39: If a be a positive real number, then show that the series
[1/2] + [1+a/2+a] + [1+2a/2+2a] + ... + [1+(n-1)a/2+(n-1)a] + ... is divergent.  [Kanpur B.Sc. 1997]
Ques40: Test the nature of the following serie :
1/3 + 1.2/3.5 + 1.2.3/3.5.7 + 1.2.3.4/3.5.7.9 + ... ... .  [Meerut 2014]
Ques41: Test the convergence of the series whose n th term is given by cos (1/n).  [Kanpur B.Sc. 1995, 2007, 13]
Ques42: Test the convergence of the series whose n th term is :
(i)  1/n log n+1/n.
(ii) 1/n sin 1/n.  [Kanpur B.Sc. 2005]
Ques43: Test the convergence of the series whose n th term is :
(i) sin (1/n).  [Kanpur B.Sc. 1996, 2012; Rohil. 2013]
(ii) n(1-cos a/n).  [Avadh 1997, 2013]
Ques44: Test the nature of the following series :
(i) 1/1.2.3 + 3/2.3.4 + 5/3.4.5 + ... .  [Avadh 2011; Agra]
(ii) 1/1√2 + 2/1 + 2√3 + 3/1 + 3√4 + ... .  [Avadh 1994]
(iii) √1/1+√1 + √2/2+√2 + √3/3+√3 + ... .  [Kanpur B.Sc. 2013, 14]
Ques45: Show that the following serie is  convergent :
1/(m+1)(m+2) + 1/(m+2)(m+3) + 1/(m+3)(m+4) + ... .  [Kanpur B.Sc. 1995]
Ques46: Find whether the following series is convergent or divergent :
1 + a + a(a+1)/1.2 + a(a+1)(a+2)/1.2.3 + ... .  [Kanpur B.Sc. 1994; Kashi 2012; Bundel. 2010]
Ques47: Test the convergence of the series whose general term is :
1 / n log n (log log n).  [Kanpur B.Sc. 1994]
Ques48: Prove that every absolutely convergent series is convergent. But the converse need not be true.  [Kanpur B.Sc. 1992, 97; Purvanchal 2010]
Ques49: Test for convergence and absolute convergence :
log(2/1) - log(3/2) + log(4/3) - log(5/4) + ... . [Kanpur B.Sc. 1997, 2013, 14; Agra 1995, 2001]
Ques50: Define absolute convergence and give one example.  [Kanpur B.Sc. 1994, 2006, 07]
Ques51: Define semi-convergent series and give one example.  [Kanpur B.Sc. 1993, 2007]
Ques52: Define absolute and conditional convergence of a series. Prove that an absolute convergent series is convergent also. Give an example to show that the converse is not true.  [Kanpur B.Sc. 1992; Purvanchal 2010]
Ques53: Prove that the series :
1-1/2+1/3-1/4+1/5-1/6+ ..., is convergent but is not absolutely convergent.  [Avadh 2012]
Ques54: Discuss the convergence of a geometric series.  [Bundel. 2008]