Indian Institute of Space Science and
Technology
MA111 - Calculus
Assignment -I
Submission deadline: 2/1/2022
1. (a) Write the ε−Ndefinition of convergence of a sequence {fn}and using this definition,
test the convergence of the sequence n2+ 3n+ 5
2n2+ 5n+ 7.
(b) Prove the limit lim
n→∞
c
np= 0,where c6= 0, p > 0 are constants, using ε−Ndefinition.
(Hint N=|c|
ε1/p )
2. (a) Find the limit lim
n→∞
8n+ (ln n)10 +n!
n6−n!(Ans: -1)
(b) Test the convergence of the sequence: xn=
n
X
k=1
3k2+ 2k
2k(Hint: it is a sequence of
partial sum of a series)
3. For a fixed positive integer m, find the values of x∈Rsuch that
lim
n→∞
m(m−1)(m−2) ·· ·(m−n+ 1)
n!xn= 0.
4. (a) Show that the sequence f(n) defined by f(1) = √2, f(n+ 1) = p2f(n) converges to
2.
(b) Verify the convergence or divergence of the sequence xn=1
n+1 +1
n+2 +1
n+3 +·· ·+1
2n, n ∈
N.(Take the difference of n and n+1 terms to prove the monotone increasing. Show
that bounded above by 1 and limit is ln 2.)
5. For what values of xthe following series converges and diverges?
(i)
∞
X
k=1
k3xk
3k(ii)
∞
X
k=1 cos(kx)
k3+3 sin(kx)
k2.(Hint: (i) Apply root test for absolute
convergence (ii) Try with comparison test )
6. Determine the convergence or divergence of the series
∞
X
n=1
n2+ 2n+ 1
n4+n2+ 2n+ 1.Justify your
answer.
7. For what values of a, the series
∞
X
n=1 a
n+ 2 −1
n+ 4converge (or) diverge.
