Lecture 6: Series
Department of Mathematics
Indian Institute of Technology Guwahati
Jul – Nov 2025
Instructor: Rajen Kumar Sinha
Series
Convergence criteria
∞
P ∞
P
Algebraic operations on series: Let xn and yn be
n=1 n=1
convergent with sums x and y respectively.
Convergence criteria
∞
P ∞
P
Algebraic operations on series: Let xn and yn be
n=1 n=1
convergent with sums x and y respectively.
Then
∞
P
1. (xn + yn ) is convergent with sum x + y
n=1
Convergence criteria
∞
P ∞
P
Algebraic operations on series: Let xn and yn be
n=1 n=1
convergent with sums x and y respectively.
Then
∞
P
1. (xn + yn ) is convergent with sum x + y
n=1
P∞
2. αxn is convergent with sum αx, where α ∈ R
n=1
Convergence criteria
∞
P ∞
P
Algebraic operations on series: Let xn and yn be
n=1 n=1
convergent with sums x and y respectively.
Then
∞
P
1. (xn + yn ) is convergent with sum x + y
n=1
P∞
2. αxn is convergent with sum αx, where α ∈ R
n=1
Cauchy criterion and Monotone sequence criterion for series
Convergence criteria
∞
P ∞
P
Algebraic operations on series: Let xn and yn be
n=1 n=1
convergent with sums x and y respectively.
Then
∞
P
1. (xn + yn ) is convergent with sum x + y
n=1
P∞
2. αxn is convergent with sum αx, where α ∈ R
n=1
Cauchy criterion and Monotone sequence criterion for series
∞
P 1
Example: n is divergent.
n=1
Necessary condition for convergence
∞
P
Result: If xn is convergent, then xn → 0.
n=1
Necessary condition for convergence
∞
P
Result: If xn is convergent, then xn → 0.
n=1
∞
P
Hence if xn 6→ 0, then xn cannot be convergent.
n=1
Necessary condition for convergence
∞
P
Result: If xn is convergent, then xn → 0.
n=1
∞
P
Hence if xn 6→ 0, then xn cannot be convergent.
n=1
Examples: The following series are not convergent.
∞ ∞
n2 +1 n
(−1)n n+2
P P
(i) (n+3)(n+4) (ii)
n=1 n=1
Test for convergence
Comparison test: Let (xn ) and (yn ) be sequences in R such that
for some n0 ∈ N, 0 ≤ xn ≤ yn for all n ≥ n0 .
Test for convergence
Comparison test: Let (xn ) and (yn ) be sequences in R such that
for some n0 ∈ N, 0 ≤ xn ≤ yn for all n ≥ n0 .
Then
∞
P P∞
(i) yn is convergent ⇒ xn is convergent.
n=1 n=1
P∞ ∞
P
(ii) xn is divergent ⇒ yn is divergent.
n=1 n=1
Test for convergence
Comparison test: Let (xn ) and (yn ) be sequences in R such that
for some n0 ∈ N, 0 ≤ xn ≤ yn for all n ≥ n0 .
Then
∞
P P∞
(i) yn is convergent ⇒ xn is convergent.
n=1 n=1
P∞ ∞
P
(ii) xn is divergent ⇒ yn is divergent.
n=1 n=1
Limit comparison test: Let (xn ) and (yn ) be sequences of positive
real numbers such that xynn → ` ∈ R.
Test for convergence
Comparison test: Let (xn ) and (yn ) be sequences in R such that
for some n0 ∈ N, 0 ≤ xn ≤ yn for all n ≥ n0 .
Then
∞
P P∞
(i) yn is convergent ⇒ xn is convergent.
n=1 n=1
P∞ ∞
P
(ii) xn is divergent ⇒ yn is divergent.
n=1 n=1
Limit comparison test: Let (xn ) and (yn ) be sequences of positive
real numbers such that xynn → ` ∈ R.
∞
P ∞
P
(i) If ` 6= 0, then xn is convergent iff yn is convergent.
n=1 n=1
P∞ ∞
P
(ii) If ` = 0, then yn is convergent ⇒ xn is convergent.
n=1 n=1
Condensation and integral tests
Cauchy’s condensation test: Let (xn ) be a decreasing sequence of
∞ ∞
2n x2n
P P
nonnegative real numbers. Then xn is convergent iff
n=1 n=1
is convergent.
Condensation and integral tests
Cauchy’s condensation test: Let (xn ) be a decreasing sequence of
∞ ∞
2n x2n
P P
nonnegative real numbers. Then xn is convergent iff
n=1 n=1
is convergent.
Examples:
∞
P 1
1. p-series: np is convergent iff p > 1.
n=1
Condensation and integral tests
Cauchy’s condensation test: Let (xn ) be a decreasing sequence of
∞ ∞
2n x2n
P P
nonnegative real numbers. Then xn is convergent iff
n=1 n=1
is convergent.
Examples:
∞
P 1
1. p-series: np is convergent iff p > 1.
n=1
∞
P 1
2. n(log n)p is convergent iff p > 1.
n=2
Condensation and integral tests
Cauchy’s condensation test: Let (xn ) be a decreasing sequence of
∞ ∞
2n x2n
P P
nonnegative real numbers. Then xn is convergent iff
n=1 n=1
is convergent.
Examples:
∞
P 1
1. p-series: np is convergent iff p > 1.
n=1
∞
P 1
2. n(log n)p is convergent iff p > 1.
n=2
Integral Test: Let f : [1, ∞) → R be monotone P∞decreasing and
f (t) ≥ 0 for all t ∈ R[1, ∞). Then the series n=1 f (n) converges if
n
and only if limn→∞ 1 f (t)dt exists.
Condensation and integral tests
Cauchy’s condensation test: Let (xn ) be a decreasing sequence of
∞ ∞
2n x2n
P P
nonnegative real numbers. Then xn is convergent iff
n=1 n=1
is convergent.
Examples:
∞
P 1
1. p-series: np is convergent iff p > 1.
n=1
∞
P 1
2. n(log n)p is convergent iff p > 1.
n=2
Integral Test: Let f : [1, ∞) → R be monotone P∞decreasing and
f (t) ≥ 0 for all t ∈ R[1, ∞). Then the series n=1 f (n) converges if
n
and only if limn→∞ 1 f (t)dt exists.
∞
P 1
Example: p-series: np is convergent iff p > 1.
n=1
