Lecture 5: Series
Department of Mathematics
Indian Institute of Technology Guwahati
Jul – Nov 2025
Instructor: Rajen Kumar Sinha
Series
Plan
Series and their convergence
Convergence criteria
Plan
Series and their convergence
Convergence criteria
Absolute convergence
Test for absolute convergence
Plan
Series and their convergence
Convergence criteria
Absolute convergence
Test for absolute convergence
Conditional convergence
Test for conditional convergence
Convergence of series
∞
P
An infinite series in R is an expression xn ,
n=1
where (xn ) is a sequence in R.
Convergence of series
∞
P
An infinite series in R is an expression xn ,
n=1
where (xn ) is a sequence in R.
More formally, it is an ordered pair ((xn ), (sn )), where (xn ) is a
sequence in R and sn = x1 + · · · + xn for all n ∈ N.
Convergence of series
∞
P
An infinite series in R is an expression xn ,
n=1
where (xn ) is a sequence in R.
More formally, it is an ordered pair ((xn ), (sn )), where (xn ) is a
sequence in R and sn = x1 + · · · + xn for all n ∈ N.
xn : nth term of the series
sn : nth partial sum of the series
Convergence of series
∞
P
An infinite series in R is an expression xn ,
n=1
where (xn ) is a sequence in R.
More formally, it is an ordered pair ((xn ), (sn )), where (xn ) is a
sequence in R and sn = x1 + · · · + xn for all n ∈ N.
xn : nth term of the series
sn : nth partial sum of the series
∞
P
Convergence of series: xn is convergent if (sn ) is
n=1
convergent.
P∞
Otherwise xn is divergent (not convergent).
n=1
Examples
∞
P
Sum of a convergent series: xn = limn→∞ sn
n=1
Examples
∞
P
Sum of a convergent series: xn = limn→∞ sn
n=1
Examples:
∞
ar n−1 (where a 6= 0) converges iff
P
1. The geometric series
n=1
|r | < 1.
Examples
∞
P
Sum of a convergent series: xn = limn→∞ sn
n=1
Examples:
∞
ar n−1 (where a 6= 0) converges iff
P
1. The geometric series
n=1
|r | < 1.
∞
1
P
2. The series n(n+1) is convergent.
n=1
Examples
∞
P
Sum of a convergent series: xn = limn→∞ sn
n=1
Examples:
∞
ar n−1 (where a 6= 0) converges iff
P
1. The geometric series
n=1
|r | < 1.
∞
1
P
2. The series n(n+1) is convergent.
n=1
3. The series 1 − 1 + 1 − 1 + · · · is not convergent.
Examples
∞
P
Sum of a convergent series: xn = limn→∞ sn
n=1
Examples:
∞
ar n−1 (where a 6= 0) converges iff
P
1. The geometric series
n=1
|r | < 1.
∞
1
P
2. The series n(n+1) is convergent.
n=1
3. The series 1 − 1 + 1 − 1 + · · · is not convergent.
Ex. If a, b ∈ R, show that the series
a + (a + b) + (a + 2b) + · · · is not convergent unless
a = b = 0.
