Infinite Series
• Sequences and Series
• Convergence of Infinite Series
• Tests of Convergence
• P-Series Test
• Comparison Tests
• Comparison of Ratio
D’alemberts ratio test
• Alternating series
• Integral test
• Leibnitz’s test
• Absolute Convergence and Conditional convergence
• Power series and Interval of convergence
• Convergence of exponential , logarithmic and binomial series
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1.1 Sequence
1.1. An infinite sequence of real numbers is an ordered unending list
of real numbers. E.g.:
1, 2, 3, 4,. . .
If f(n) = an , the sequence is
written as a1, a2 , a and denoted by , an
... n Here f(n) or an are the
th
n terms of the Sequence.
Ex. 1. 1 , 4 , 9 , 16 ,......... n2 ,. ... (or) n2
N S
1
2 4
3 9
. .
. .
n n2
. .
. .
1 1 1 1 1
Ex. 2. , 3, 3,..... ....(or) 3
n
3 3
1 2 3 n
Ex. 3. 1, 1, 1......1..... or <1>
3
( −1)
n −1
Ex 4: 1 , –1, 1, –1, ......... or
Note : 1. If S R then the sequence is called a real sequence.
2. The range of a sequence is almost a countable set.
1.1.1 Kinds of Sequences
1. Finite Sequence: A sequence an in which n--→infinity an = l is said to
(limit )
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be a finite Sequence. i.e., A finite Sequence has a finite number of terms.
2. Infinite Sequence: A sequence, which is not finite, is an infinite sequence.
Limits of a Sequence
A Sequence an is said to tend to limit ‘l’ when, given any + ve number ' ',
Then we
write Lt a n= l or an → l
n→
Monotonic Sequences.
A monotonic sequence is a sequence that always increases or always
decreases. For instance, 1/n is a monotonic decreasing sequence, and
an +1 >= an
an +1 <= an
n = 1, 2, 3, 4, . . . is a monotonic increasing sequence.
1 , 4 7, 10 --------------------
1,1/2 , 1/3 , ¼ -----------------------
1-1 , 1, -1 ,1 -----------------------
Convergent, Divergent and Oscillatory Sequences
a) Convergent Sequence: A sequence which tends to a finite limit, say ‘l’ is
called a Convergent Sequence. We say that the sequence converges to ‘l’
Eg : 1 , ¼ , 1/9 ----------
b) Divergent Sequence: A sequence which tends to is said to be Divergent
(or is said to diverge).
Eg: 3 , 5, 7, ------------
c) Oscillatory Sequence: A sequence which neither converges nor diverges ,is
called an Oscillatory Sequence.
Eg: 1, -1 ,1 ,– 1 ----------------------
EG : a) 1,3,5,7 , -----------------
b) 1 , ½ ,1/3 , -------------------
3 4 5
Ex. 1. Consider the sequence 2 , , , ,...... here
2 3 4
1
a =1+ 3
n
n
The sequence an is convergent and has the limit 1
1
If an = 3 + ( −1)
n
Ex. 2.
'n '
an converges to 3.
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If an = n 2 + ( −1) .n, an diverges.
n
Ex. 3.
1
If a = + 2 ( −1) ,
n
Ex. 4.
n
n
Arithmetic Progression , Tn = a + (n-1)d
Sn = n/2 [2a + (n-1) d]
Geometric Series
he series, 1 + x + x2 + .....xn−1 + ... is
(i) Sn = a(1-rn) /(1-r) r<1,
when
11) Sn = a(rn- -1 ) /(r-1) when r 1 .
Infinite Series
If un is a sequence, then the expression u1 + u2 + u3 + ........ + un +is called an
infinite series. It is denoted by un or simply un
n=1
The sum of the first n terms of the series is denoted by sn
i.e., sn = u1 + u2 + u3 + ...... + un ; s1 , s2 , s3 ,....sn are called partial sums.
Convergent, Divergent and Oscillatory Series
Let un be an infinite series. As n → , there are three possibilities.
(a) Convergent series: As n → , sn → a finite limit, say ‘s’ in which case the
series is said to be convergent and ‘s’ is called its sum to infinity.
Thus Lt s n= s n
n→
This is also written as u1 + u2 + u3 + ..... + un + ...to = s. (or) un= s (or)
n=1
simply un = s.
(b) Divergent series: If sn → or − , the series said to be divergent.
(c) Oscillatory Series: If sn does not tend to a unique limit either finite or infinite it
is said to be an Oscillatory Series.
Note: Divergent or Oscillatory series are sometimes called non convergent series.
Some General Properties of Infinite Series
1. The convergence or divergence of an infinites series is unaltered by an addition or
deletion of a finite number of terms from it.
2. Let un converge to ‘s’
Let ‘k’ be a non – zero fixed number. Then kun converges to ks.
Also, if un diverges or oscillates, so does kun
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