An infinite series in R is an expression
∞
P
n=1
where (xn ) is a sequence in R.
xn ,
An infinite series in R is an expression
∞
P
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.
An infinite series in R is an expression
∞
P
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
An infinite series in R is an expression
∞
P
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
n=1
xn is convergent if (sn ) is
convergent.
∞
P
Otherwise
xn is divergent (not convergent).
n=1
Sum of a convergent series:
∞
P
n=1
xn = lim sn
n→∞
Sum of a convergent series:
∞
P
n=1
xn = lim sn
n→∞
Examples:
1. The geometric series
∞
P
n=1
|r | < 1.
ar n−1 (where a 6= 0) converges iff
Sum of a convergent series:
∞
P
xn = lim sn
n=1
n→∞
Examples:
1. The geometric series
∞
P
n=1
|r | < 1.
2. The series
∞
P
n=1
1
n(n+1)
ar n−1 (where a 6= 0) converges iff
is convergent.
Sum of a convergent series:
∞
P
xn = lim sn
n=1
n→∞
Examples:
1. The geometric series
∞
P
n=1
|r | < 1.
2. The series
∞
P
n=1
1
n(n+1)
ar n−1 (where a 6= 0) converges iff
is convergent.
3. The series 1 − 1 + 1 − 1 + · · · is not convergent.
Sum of a convergent series:
∞
P
xn = lim sn
n=1
n→∞
Examples:
1. The geometric series
∞
P
n=1
|r | < 1.
2. The series
∞
P
n=1
1
n(n+1)
ar n−1 (where a 6= 0) converges iff
is convergent.
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.
Algebraic operations on series: Let
∞
P
xn and
n=1
convergent with sums x and y respectively.
∞
P
n=1
yn be
Algebraic operations on series: Let
∞
P
xn and
n=1
convergent with sums x and y respectively.
Then
1.
∞
P
(xn + yn ) is convergent with sum x + y .
n=1
∞
P
n=1
yn be
Algebraic operations on series: Let
∞
P
xn and
n=1
∞
P
n=1
convergent with sums x and y respectively.
Then
1.
∞
P
(xn + yn ) is convergent with sum x + y .
n=1
2.
∞
P
n=1
αxn is convergent with sum αx , where α ∈ R.
yn be
Algebraic operations on series: Let
∞
P
xn and
n=1
∞
P
yn be
n=1
convergent with sums x and y respectively.
Then
1.
∞
P
(xn + yn ) is convergent with sum x + y .
n=1
2.
∞
P
n=1
αxn is convergent with sum αx , where α ∈ R.
Monotonic criterion: A series
is convergent iff the
∞
P
xn of non-negative terms
n=1
sequence (sn ) is bounded above.
Algebraic operations on series: Let
∞
P
xn and
n=1
∞
P
yn be
n=1
convergent with sums x and y respectively.
Then
1.
∞
P
(xn + yn ) is convergent with sum x + y .
n=1
2.
∞
P
n=1
αxn is convergent with sum αx , where α ∈ R.
Monotonic criterion: A series
is convergent iff the
Example:
∞
P
n=1
1
n2
∞
P
xn of non-negative terms
n=1
sequence (sn ) is bounded above.
is convergent.
Cauchy criterion:
∞
P
xn is convergent iff for each ε > 0, there
n=1
exists n0 ∈ N such that |xn+1 + · · · + xm | < ε for all m > n ≥ n0 .
Cauchy criterion:
∞
P
xn is convergent iff for each ε > 0, there
n=1
exists n0 ∈ N such that |xn+1 + · · · + xm | < ε for all m > n ≥ n0 .
Example:
∞
P
n=1
1
n
is not convergent.
Cauchy criterion:
∞
P
xn is convergent iff for each ε > 0, there
n=1
exists n0 ∈ N such that |xn+1 + · · · + xm | < ε for all m > n ≥ n0 .
Example:
∞
P
n=1
Result: If
∞
P
n=1
1
n
is not convergent.
xn is convergent, then xn → 0.
Cauchy criterion:
∞
P
xn is convergent iff for each ε > 0, there
n=1
exists n0 ∈ N such that |xn+1 + · · · + xm | < ε for all m > n ≥ n0 .
Example:
∞
P
n=1
Result: If
∞
P
n=1
1
n
is not convergent.
xn is convergent, then xn → 0.
Hence if xn 6→ 0, then
∞
P
n=1
xn cannot be convergent.
Cauchy criterion:
∞
P
xn is convergent iff for each ε > 0, there
n=1
exists n0 ∈ N such that |xn+1 + · · · + xm | < ε for all m > n ≥ n0 .
Example:
∞
P
n=1
Result: If
∞
P
n=1
1
n
is not convergent.
xn is convergent, then xn → 0.
Hence if xn 6→ 0, then
∞
P
xn cannot be convergent.
n=1
Examples: The following series are not convergent.
∞
∞
P
P
n2 +1
n
(i)
(ii)
(−1)n n+2
(n+3)(n+4)
n=1
n=1
Comparison test: Let (xn ) and (yn ) be sequences in R such
that for some n0 ∈ N, 0 ≤ xn ≤ yn for all n ≥ n0 .
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.
(ii)
n=1
∞
P
n=1
n=1
xn is divergent ⇒
∞
P
n=1
yn is divergent.
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.
(ii)
n=1
∞
P
n=1
n=1
xn is divergent ⇒
∞
P
yn is divergent.
n=1
Limit comparison test: Let (xn ) and (yn ) be sequences of
positive real numbers such that xynn → ` ∈ R.
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.
(ii)
n=1
∞
P
n=1
n=1
xn is divergent ⇒
∞
P
yn is divergent.
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.
(ii) If ` = 0, then
n=1
∞
P
n=1
yn is convergent ⇒
n=1
∞
P
n=1
xn is convergent.
Cauchy’s condensation test: Let (xn ) be a decreasing
∞
P
sequence of nonnegative real numbers. Then
xn is
convergent iff
∞
P
n=1
n=1
2n x2n
is convergent.
Cauchy’s condensation test: Let (xn ) be a decreasing
∞
P
sequence of nonnegative real numbers. Then
xn is
convergent iff
∞
P
n=1
2n x2n
is convergent.
n=1
Examples:
1. p-series:
∞
P
n=1
1
np
is convergent iff p > 1.
Cauchy’s condensation test: Let (xn ) be a decreasing
∞
P
sequence of nonnegative real numbers. Then
xn is
convergent iff
∞
P
n=1
2n x2n
is convergent.
n=1
Examples:
1. p-series:
∞
P
n=1
2.
∞
P
n=2
1
n(log n)p
1
np
is convergent iff p > 1.
is convergent iff p > 1.
Ex. Examine whether the following series are convergent.
∞
∞
∞
P
P
P
1+sin n
1
√ 1
(ii)
(iii)
(i)
2
2n +n
n=1
(iv)
1+n
n=1
∞ √
√
P
( n + 1 − n)
n=1
n=2
(v)
∞
P
n=1
1
n!
n(n−1)
(vi)
∞
P
n=1
n
4n3 −2
Ex. Examine whether the following series are convergent.
∞
∞
∞
P
P
P
1+sin n
1
√ 1
(ii)
(iii)
(i)
2
2n +n
n=1
(iv)
1+n
n=1
∞ √
√
P
( n + 1 − n)
n=1
Definitions:
convergent.
∞
P
n=1
n=2
(v)
∞
P
n=1
1
n!
n(n−1)
(vi)
∞
P
n=1
n
4n3 −2
xn is called absolutely convergent if
∞
P
n=1
|xn | is
Ex. Examine whether the following series are convergent.
∞
∞
∞
P
P
P
1+sin n
1
√ 1
(ii)
(iii)
(i)
2
2n +n
n=1
(iv)
1+n
n=1
∞ √
√
P
( n + 1 − n)
n=1
Definitions:
∞
P
n=2
(v)
∞
P
n=1
1
n!
n(n−1)
(vi)
∞
P
n=1
n
4n3 −2
xn is called absolutely convergent if
n=1
∞
P
n=1
|xn | is
convergent.
∞
∞
P
P
xn is called conditionally convergent if
xn is convergent
n=1
but
∞
P
n=1
n=1
|xn | is divergent.
Result: Every absolutely convergent series is convergent.
Result: Every absolutely convergent series is convergent.
Ratio test: Let (xn ) be a sequence of nonzero real numbers
x
such that | n+1
xn | → `.
Result: Every absolutely convergent series is convergent.
Ratio test: Let (xn ) be a sequence of nonzero real numbers
x
such that | n+1
xn | → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
Result: Every absolutely convergent series is convergent.
Ratio test: Let (xn ) be a sequence of nonzero real numbers
x
such that | n+1
xn | → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
Examples: (i)
∞
P
n=1
n
2n
(ii)
∞
P
n=1
n!
nn
(iii)
∞
P
n=1
(2n)!
(n!)2
Result: Every absolutely convergent series is convergent.
Ratio test: Let (xn ) be a sequence of nonzero real numbers
x
such that | n+1
xn | → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
Examples: (i)
∞
P
n=1
n
2n
(ii)
∞
P
n=1
n!
nn
Ex. Find all real values of x for which
(iii)
∞
P
n=1
∞
P
n=1
xn
n!
(2n)!
(n!)2
converges.
1
Root test: Let (xn ) be a sequence in R such that |xn | n → `.
1
Root test: Let (xn ) be a sequence in R such that |xn | n → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
1
Root test: Let (xn ) be a sequence in R such that |xn | n → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
Examples: (i)
(iv)
∞
P
n=1
∞
P
n=1
5n
3n +4n
(n!)n
nn2
(ii)
∞
P
n=1
n n
( n+1
)
2
(iii)
∞
P
n=1
nn
2n 2
1
Root test: Let (xn ) be a sequence in R such that |xn | n → `.
(i) If ` < 1, then
∞
P
xn is absolutely convergent.
(ii) If ` > 1, then
n=1
∞
P
xn is divergent.
n=1
Examples: (i)
(iv)
∞
P
n=1
∞
P
n=1
(n!)n
nn2
(ii)
∞
P
n=1
n n
( n+1
)
2
(iii)
∞
P
n=1
nn
2n 2
5n
3n +4n
Ex. Test the convergence of the series
1 + 2x + x 2 + 2x 3 + x 4 + 2x 5 + x 6 + 2x 7 + · · · , where x ∈ R.
Leibniz’s test: Let (xn ) be a decreasing sequence of positive
real numbers such that xn → 0.
Leibniz’s test: Let (xn ) be a decreasing sequence of positive
real numbers such that xn → 0.
∞
P
Then the alternating series
(−1)n+1 xn is convergent.
n=1
Leibniz’s test: Let (xn ) be a decreasing sequence of positive
real numbers such that xn → 0.
∞
P
Then the alternating series
(−1)n+1 xn is convergent.
n=1
Examples: (i)
(iii)
∞
P
(−1)n+1 n1p , p ∈ R
n=1
√
n+1
(−1)n+1 n+1
n=1
∞
P
(ii)
∞
P
(−1)n+1 n3n+1
n=1
Leibniz’s test: Let (xn ) be a decreasing sequence of positive
real numbers such that xn → 0.
∞
P
Then the alternating series
(−1)n+1 xn is convergent.
n=1
Examples: (i)
(iii)
∞
P
(−1)n+1 n1p , p ∈ R
n=1
√
n+1
(−1)n+1 n+1
n=1
(ii)
∞
P
(−1)n+1 n3n+1
n=1
∞
P
Result: Grouping of terms of a convergent series does not
change the convergence and the sum.
Leibniz’s test: Let (xn ) be a decreasing sequence of positive
real numbers such that xn → 0.
∞
P
Then the alternating series
(−1)n+1 xn is convergent.
n=1
Examples: (i)
(iii)
∞
P
(−1)n+1 n1p , p ∈ R
n=1
√
n+1
(−1)n+1 n+1
n=1
(ii)
∞
P
(−1)n+1 n3n+1
n=1
∞
P
Result: Grouping of terms of a convergent series does not
change the convergence and the sum.
However, a divergent series can become convergent after
grouping of terms.
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Example: 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+ ··· = s
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Example: 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+ ··· = s
1+
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+ · · · = 32 s
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Example: 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+ ··· = s
1+
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+ · · · = 32 s
Riemann’s rearrangement theorem: Let
∞
P
n=1
conditionally convergent series.
xn be a
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Example: 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+ ··· = s
1+
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+ · · · = 32 s
Riemann’s rearrangement theorem: Let
∞
P
xn be a
n=1
conditionally convergent series.
(i) If s ∈ R, then there exists a rearrangement of terms of
∞
P
xn such that the rearranged series has the sum s.
n=1
Result: Rearrangement of terms does not change the
convergence and the sum of an absolutely convergent series.
Example: 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+ ··· = s
1+
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+ · · · = 32 s
Riemann’s rearrangement theorem: Let
∞
P
xn be a
n=1
conditionally convergent series.
(i) If s ∈ R, then there exists a rearrangement of terms of
∞
P
xn such that the rearranged series has the sum s.
n=1
(ii) There exists a rearrangement of terms of
∞
P
n=1
the rearranged series diverges.
xn such that
Power series: A series of the form
∞
P
n=0
an (x − x0 )n ,
where x0 , an ∈ R for n = 0, 1, 2, ... and x ∈ R.
Power series: A series of the form
∞
P
n=0
an (x − x0 )n ,
where x0 , an ∈ R for n = 0, 1, 2, ... and x ∈ R.
It is sufficient to consider the series
∞
P
n=0
an x n .
Power series: A series of the form
∞
P
n=0
an (x − x0 )n ,
where x0 , an ∈ R for n = 0, 1, 2, ... and x ∈ R.
It is sufficient to consider the series
∞
P
an x n .
n=0
Convergence - Examples:
∞ n
∞
P
P
x
(i)
(ii)
n!x n
n!
n=0
n=0
(iii)
∞
P
n=0
xn
Power series: A series of the form
∞
P
n=0
an (x − x0 )n ,
where x0 , an ∈ R for n = 0, 1, 2, ... and x ∈ R.
It is sufficient to consider the series
∞
P
an x n .
n=0
Convergence - Examples:
∞ n
∞
P
P
x
(i)
(ii)
n!x n
n!
n=0
n=0
Result:
(i) If
∞
P
n=0
(iii)
∞
P
xn
n=0
an x n converges for x = x1 6= 0, then it converges
absolutely for all x ∈ R satisfying |x| < |x1 |.
Power series: A series of the form
∞
P
n=0
an (x − x0 )n ,
where x0 , an ∈ R for n = 0, 1, 2, ... and x ∈ R.
It is sufficient to consider the series
∞
P
an x n .
n=0
Convergence - Examples:
∞ n
∞
P
P
x
(i)
(ii)
n!x n
n!
n=0
n=0
Result:
(i) If
∞
P
n=0
(iii)
∞
P
xn
n=0
an x n converges for x = x1 6= 0, then it converges
absolutely for all x ∈ R satisfying |x| < |x1 |.
∞
P
(ii) If
an x n diverges for x = x2 , then it diverges for all x ∈ R
n=0
satisfying |x| > |x2 |.
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
The series may or may not converge for |x| = R.
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
The series may or may not converge for |x| = R.
Methods to find the interval of convergence:
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
The series may or may not converge for |x| = R.
Methods to find the interval of convergence:
Ex. Find the radius of convergence of the power series
mentioned earlier.
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
The series may or may not converge for |x| = R.
Methods to find the interval of convergence:
Ex. Find the radius of convergence of the power series
mentioned earlier.
Ex. Find the interval of convergence of the following power
series.
∞ n
∞ n
∞
P
P
P
(−1)n
x
x
n
(ii)
(iii)
(i)
2
n
n·4n (x − 1)
n
n=1
n=0
n=1
Radius of convergence: For every power series
∞
P
an x n , there
n=0
exists a unique R satisfying 0 ≤ R ≤ ∞ such that the series
converges absolutely if |x| < R and diverges if |x| > R.
The series may or may not converge for |x| = R.
Methods to find the interval of convergence:
Ex. Find the radius of convergence of the power series
mentioned earlier.
Ex. Find the interval of convergence of the following power
series.
∞ n
∞ n
∞
P
P
P
(−1)n
x
x
n
(ii)
(iii)
(i)
2
n
n·4n (x − 1)
n
n=1
n=0
n=1
Term-by-term operations on power series: