Calculus For Engineers
Sequence and Series
Dr. Ramesh Arumugam
Department of Mathematics
School of Advanced Sciences
VIT-AP University, Amaravati
ramesh.a@vitap.ac.in
Sequence of Real numbers
DEFINITION: A sequence of real numbers is a function defined on
the set N = {1, 2, ...} of natural numbers whose range is contained in
the set R of real numbers.
X:N→R denoted by (xn ) or {xn }
Examples:
Convergence and Divergence
Convergence and Divergence
Check the divergence of following the sequences {an }:
n2 − 2n + 1
(a) an =
n−1
2n + 1
(b) an = √
1−3 n
Examples
Which of the sequences {an } converge?
1 − 2n
(a) an =
1 + 2n
√
n
(b) an = 10n
1
(c) an = n 1 − cos
n
√ 1
(d) an = n sin √
n
The Sandwich Theorem for Sequences
The Sandwich Theorem
Let {an }, {bn } and {cn } be sequences of real numbers. If an ≤ bn ≤ cn
holds for all n beyond some index N , and if limn→∞ an = limn→∞ cn =
L, then
lim bn = L.
n→∞
The Sandwich Theorem for Sequences
The Sandwich Theorem
Let {an }, {bn } and {cn } be sequences of real numbers. If an ≤ bn ≤ cn
holds for all n beyond some index N , and if limn→∞ an = limn→∞ cn =
L, then
lim bn = L.
n→∞
Examples:
Bounded Sequences
Bounded sequence
The sequence {an } is bounded below if there exists m ∈ R such
that m ≤ an for all n ∈ N. The number m is called a lower
bound.
The sequence {an } is bounded above if there exists M ∈ R such
that an ≤ M for all n ∈ N. The number M is called a upper
bound.
The sequence {an } is bounded if an is bounded above and below.
i.e. there exists M, m ∈ R such that m ≤ an ≤ M for all n ∈ N.
Examples:
1 n
n o n o
{n} , , , {cos n}
n n+1
Monotonic Sequences
Monotonic sequence
The sequence {an } is increasing if it satisfies the inequalities
a1 ≤ a2 ≤ . . . ≤ an ≤ an+1 ≤ . . .
The sequence {an } is decreasing if it satisfies the inequalities
a1 ≥ a2 ≥ . . . ≥ an ≥ an+1 ≥ . . .
The sequence {an } is monotone if it is either increasing or
decreasing
Check the follwing sequences:
(1, 2, 3, ..., n, ...) (1, 1/2, 1/3, ..., 1/n, ...)
(1, −1, 1, −1, ..., (−1)n+1 , ...) (1, 2, 2, 3, 3, 3, ...)
(1, 1/2, 1/22 , ..., 1/2n−1 , ...) (b, b2 , b3 , ..., bn , ...), b > 1
n
2 3
(a, a , a , ..., a , ...), a < 1 (−1, 2, −3, 4, ..., (−1)n n, ...)
Monotone Convergence Theorem
Monotone Convergence Theorem
If a sequence {an } is both bounded and monotonic, then the sequence
converges.
Check:
(1, −1, 1, −1, ..., (−1)n+1 , ...)
(1, 1/2, 1/22 , ..., 1/2n−1 , ...)
(1, 1/2, 1/3, ..., 1/n, ...)
(−1, 2, −3, 4, ..., (−1)n n, ...)
a1 = 1, an+1 = an + ( 15 )n
Series
Geometric Series
Geometric series are series of the form
∞
X
a + ar + ar2 + . . . + arn−1 + . . . = arn−1 , a 6= 0
n=1
Geometric Series
Geometric series are series of the form
∞
X
a + ar + ar2 + . . . + arn−1 + . . . = arn−1 , a 6= 0
n=1
Geometric Series
Geometric series are series of the form
∞
X
a + ar + ar2 + . . . + arn−1 + . . . = arn−1 , a 6= 0
n=1
Convergence of a Series
Theorem
∞
X
an converges, then an → 0
n=1
Convergence of a Series
Theorem
∞
X
an converges, then an → 0
n=1
The nth-Term Test for Divergence
∞
X
an diverges if lim an fails to exist or is different from zero.
n→∞
n=1
Convergence of a Series
Theorem
∞
X
an converges, then an → 0
n=1
The nth-Term Test for Divergence
∞
X
an diverges if lim an fails to exist or is different from zero.
n→∞
n=1
p-series
∞
X 1
The p − series converges if p > 1, diverges if p ≤ 1.
np
n=1
Convergence Tests
Direct Comparison Test
P∞ P∞
Let n=1
an and n=1
bn be two series with 0 ≤ an ≤ bn for all n.
Then
P P
If bn converges, then an also converges
P P
If an diverges, then an also diverges
Examples:
∞ ∞ ∞ ∞
X 1 X n−1 X cos n2 X n+2
(i) , (ii) , (iii) , (iv)
n2 + 30 n4 + 2 n3/2 n2 − n
n=1 n=2 1 n=2
Convergence Tests
Limit Comparison Test
Suppose an > 0 and bn > 0 for all n ≥ N .
an X X
1. If lim = c and c > 0, then an and bn both converge or
n→∞ bn
both diverge.
an X X
2. If = 0 and
lim bn converges, then an converges.
n→∞ bn
an X X
3. If lim = ∞ and bn diverges, then an diverges.
n→∞ bn
Examples:
∞ ∞ ∞
X n−2 X 1 X (n + 1) 1
(i) , (ii) , (iii)
n3 − n2 + 3 ln n (n2 + 1) 5n
n=1 n=2 1
Convergence Tests
Absolute Convergence Test
∞
X ∞
X
If |an | converges, then an converges.
n=1 n=1
Ratio Test
X an+1
Let an be any series and suppose that lim = ρ. Then
n→∞ an
(a) the series converges absolutely if ρ < 1
(b) the series diverges if ρ > 1
(c) the test is inconclusive if ρ = 1.
Check the convergence of the series:
∞ ∞ ∞
X 2n X 3n+1 X 2n2 + 5
(i) , (ii) , (iii)
n! ln n n4 + 1
n=1 n=2 1
Convergence Tests
Root Test
X p
n
Let an be any series and suppose that lim |an | = ρ. Then
n→∞
(a) the series converges absolutely if ρ < 1
(b) the series diverges if ρ > 1
(c) the test is inconclusive if ρ = 1.
Check:
∞ ∞ ∞
X 4n X n 1 X 1
(i) n
, (ii) sin √ , (iii)
(3n) n np
n=1 n=1 1
Convergence Test
Example
A rubber ball when dropped on a hard floor bounces back up to the 34
of the height from which it was dropped. We decide to ignore effects
of elasticity and small-scale phenomena. Find the distance that the ball
travels if it is dropped from the height of 64 meters.
Convergence Test
Example
A rubber ball when dropped on a hard floor bounces back up to the 34
of the height from which it was dropped. We decide to ignore effects
of elasticity and small-scale phenomena. Find the distance that the ball
travels if it is dropped from the height of 64 meters.
