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Assignment 5

The document is an assignment from the Department of Mathematics at Visvesvaraya National Institute of Technology, Nagpur, focusing on sequences, series, and improper integrals. It includes various problems related to the convergence of sequences and series, as well as the examination of improper integrals. The assignment requires students to prove convergence, find limits, and analyze the behavior of given mathematical expressions.

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10 views2 pages

Assignment 5

The document is an assignment from the Department of Mathematics at Visvesvaraya National Institute of Technology, Nagpur, focusing on sequences, series, and improper integrals. It includes various problems related to the convergence of sequences and series, as well as the examination of improper integrals. The assignment requires students to prove convergence, find limits, and analyze the behavior of given mathematical expressions.

Uploaded by

gameboy10t
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Visvesvaraya National Institute of Technology, Nagpur

Department of Mathematics 1
Mathematics-I (MAL-101)
Assignment on sequence, series, improper integrals

Sequences:
1. Discuss the convergence or otherwise of the following sequences.

() ((2n)(vi)
n!n! +2
2. Show that the sequence {r"} (i) diverges to +oo if r > 1, (ii) converges to 1 if r = 1, (ii)
converges to 0 if -1 <r<1,(iv) is oscillating finitely if r = -1, (v) is oscillating infinitely

ifr<-1.

3. Prove that the sequence {an} defined by a1 = v2, dn+1 = V24n for all n 1, is convergent
and the limit of the sequence is 2.

4. Using Sandwich theorem, prove that 1


?+1 n +2 Vn2+n
5. Prove that the sequence {a,} defined by aj = V7, an+1 = V7 +an for all n, is convergent
and the limit of the sequence is the positive root of the equation z - t -7= 0.

6. Prove that the sequence {an} defined by a1 = 0, a2 = 1. and an+2 = (an+1 +an) for all
n1, is a Cauchy sequence.

Infinite Series:

7. Examine the behaviour of the following series. Also find their sum, if exists.

2-1
n=l n=l
i-1)"1
6-1
n=l| n=0

8. Examine the behaviour of the following series (converges or diverges):

2n-1 n!2"
.. (V) 2 n
nn+)(n 1)(n +2) +Vn+11 n=l

) )G)G9
12 12.22 12.32.52 2! 3!
1
(V1) 2 2 2 t 6 t (ix) 1 +5+ + (x) P> 0
2 (lnn)p
T=2

(-1)"-n (xi) 1 +2) +1+2+3)1 +2+3+4) +


-

n=
Power Series:
1
9. Find the radius and interval of the convergence of the following power series:
(n +1) 1.3 2 1.3.53 (ii) +3)"
(n +2)(n +3)"
n=l
2T 25t 25+ n=l

v) 4r-8)".
8
n=l n=l

10. Find the radius and the circle of the convergence the following power series:
n2
3n + o" Gi) (GV)
2 (2n)!
n=0 n=0

Improper Inte rals

11. Examine the convergence of the following integrals:

dz
o ) ar n hr (ii)log,(sin z) dr

da
iv) Jo cos a - cos t
(

dr(vi) dr (vi) 1+:


d

da Sin a da
(ii
+ ] ), zydr ()(1+z)2+ z)Vz(1 -

r)

(i) Inz dz.

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