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TOPIC:SEQUENCE AND SERIES

SUBJECT: ENGINEERING MATHEMATICS - SUBJECT CODE: MTH - 101

 

SUBMITTED TO:- SUBMITTED BY:-

Miss. Sofia Singla Name: Jagraj Singh

(Dept. of Maths.) Regd. No: 10807772

Section: 211

Roll No: R211A23Course: B.Tech-MBA EC

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ACKNOWLEDGEMENT

I Jagraj Singh of section 211, registration no. 10807772 and rno. R211A23 of B.Tech – MBA (ECE) hereby submit thterm paper of Engineering Mathematics – I on the topSequence and Series to Miss. Sofia Singla (Dept. Mathematics.). It’s my pleasure to submit my expresseviews on this topic.

 I also want to thanks to Lovely profession

university to giving me this enlarged opportunity.

 Submitted to:Miss. Sofia Singla(Dept. of Maths.)

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Contents

• 1 Examples and notation

• 2 Types and properties of sequences

• 3 Sequences in analysis

• 4 Series

• 5 Infinite sequences in theoretical computer science

• 6 Sequences as vectors

• 7 Doubly-infinite sequences

• 8 Ordinal-indexed sequence

• 9 Sequences and automata

Examples and notation

There are various and quite different notions of sequences in mathematics, some of which ( e.g., exact sequeare not covered by the notations introduced below.

A sequence may be denoted (a1, a2, ...). For shortness, the notation (an) is also used.

A more formal definition of a finite sequence with terms in a set S is a function from {1, 2, ..., n} to S for s

n ≥ 0. An infinite sequence in S is a function from {1, 2, ...} (the set of natural numbers without 0) to S .

Sequences may also start from 0, so the first term in the sequence is then a0.

A sequence of a fixed-length n is also called an n-tuple . Finite sequences include the empty sequence ( ) that

no elements.

A function from all integers into a set is sometimes called a bi-infinite sequence, since it may be thought o

a sequence indexed by negative integers grafted onto a sequence indexed by positive integers.

Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of

elements without disturbing the relative positions of the remaining elements.

If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one

which each term is greater than or equal to the term before it; if each term is strictly greater than the  preceding it, the sequence is called  strictly monotonically increasing . A monotonically decreasing sequenc

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defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This

special case of the more general notion of monotonic function.

The terms non-decreasing and non-increasing are used in order to avoid any possible confusion with stri

increasing and strictly decreasing, respectively. If the terms of a sequence are integers, then the sequence i

integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence.

If S is endowed with a topology, then it becomes possible to consider convergence of an infinite sequence i

Such considerations involve the concept of the limit of a sequence.

Sequences in analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

or 

which is to say, infinite sequences of elements indexed by natural numbers.

It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequedefined by  xn = 1/log(n) would be defined only for  n ≥ 2. When talking about such infinite sequences,

usually sufficient (and does not change much for most considerations) to assume that the members of

sequence are defined at least for all indices large enough, that is, greater than some given N .)

The most elementary type of sequences are numerical ones, that is, sequences of real or  complex numbers. type can be generalized to sequences of elements of some vector space. In analysis, the vector sp

considered are often function spaces. Even more generally, one can study sequences with elements in so

topological space.

Series

 Main article: Series (mathematics)

The sum of terms of a sequence is a series. More precisely, if ( x1, x2, x3, ...) is a sequence, one may consider

sequence of  partial sums (S 1, S 2, S 3, ...), with

Formally, this pair of sequences comprises the series with the terms x1, x2, x3, ..., which is denoted as

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For mdetails, see series.

Infinite sequences in theoretical computer science

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theore

computer science. They are often referred to simply as  sequences (as opposed to finite  strings). Infinite binsequences, for instance, are infinite sequences of  bits (characters drawn from the alphabet {0,1}). The set

{0, 1}∞ of all infinite, binary sequences is sometimes called the Cantor space.

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An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of

sequence to 1 if and only if the n th string (in shortlex order ) is in the language. Therefore, the stud

complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences.

An infinite sequence drawn from the alphabet {0, 1, ..., b−1} may also represent a real number expressed in

 base-b  positional number system. This equivalence is often used to bring the techniques of  real analysis to on complexity classes.

Sequences as vectors

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of  F -va

sequences (where F is a field) is a function space (in fact, a product space) of  F -valued functions over the senatural numbers.

In particular, the term  sequence space usually refers to a linear subspace of the set of all possible infi

sequences with elements in .

Doubly-infinite sequences

 Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in

other -- the sequence has a first element, but no final element (a  singly-infinite sequence). A doubly-infi

 sequence is infinite in both directions -- it has neither a first nor a final element. Singly-infinite sequencesfunctions from the natural numbers (N') to some set, whereas doubly-infinite sequences are functions from

integers (Z) to some set.

One can interpret singly infinite sequences as element of the semigroup ring of the natural numbers , and dou

infinite sequences as elements of the group ring of the integers . This perspective is used in the Cauchy proof sequences.

Ordinal-indexed sequence

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an

indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence iordinary sequence.

Sequences and automata

Automata or finite state machines can typically thought of as directed graphs, with edges labeled using so

specific alphabet Σ. Most familiar types of automata transition from state to state by reading input letters f

Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence callword (or input word). The sequence of states encountered by the automaton when processing a word is call

run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more tone successor for some input letter. This is typically thought of as producing multiple possible runs for a gi

word, each being a sequence of single states, rather than producing a single run that is a sequence of set

states; however, 'run' is occasionally used to mean the latter.

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Calculating specific terms leads to an "nth term

Examples:

1. For  {0, 1, 2, 3, 4} and {0, 1, 2, 3, 4, 5, . . . } , the formula

. But this is unhandy since we need the previous term

is easier to notice that . This is the nth term formula.

Where's the difference in the two sequences? In the domain of sequences. An infinite sequence is understood to have the domain of natural or counting numbers. (Sometimes the whole numbers are us

when it is convenient.) What we are doing is setting up a one-to-ocorrespondence between the set of natural numbers and an ordered lisvalues (the terms).

2. For {10, 11, 12, 13, 14, . . .}, the formula is .3. For {1, 4, 9, 16, 25, . . . , 100}, the formula is .4. The following sequence is very different: {1, 1, 2, 3, 5, 8, 13, ...}. This is

Fibonacci  sequence. In this case we are forced to write a formula thexplains the next term in terms of the previous two terms.

Before I go further, see if you can analyze the process creating terms.

Now see if you can complete the formula. This one has to set the ftwo terms specifically, so a1 = 1 and a2 = 1. Then an =

A curious fact about infinite sets. Analyzing their sizes can lead interesting results. Sets 1 and 2 above are exactly the same size!

No way, you say? Set 2 has the number zero and Set 1 does not. Well, think about t

function  .

The x values come from the the sequence 0, 1, 2, 3, 4, 5, . . . and the y values are tsequence 1, 2, 3, 4, 5, . .

It is easy to see that we can match up every x with exactly one y . This is called a onto-one correspondence. Finding it is enough to prove the sizes are the same.

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More About Notations

In function notation, to ask for a value at 1 in a discrete function, we wouldwrite f  (1). In sequence notation we write a1. The two ways of asking for avalue produce exactly the same result when we are 5.

AREThmetic Sequences Series 

Arithmetic Sequences 

Exercise: Find the next term and the general formula for the following:

 A. {2, 5, 8, 11, 14,

B. {0, 4, 8, 12, 16,

C. {2, -1, -4, -7, -10,

 

For each of these three sequences there is a common difference. In the first sequethe common difference is d = 3, in the second sequence the common difference is d4, and on the third sequence the common difference is d = -3. We will call a sequean arithmetic sequence if there is a common difference.

The general formula for an arithmetic sequence is

an = a1 + (n - 1)d 

Example 

What is the difference between the fourth and the tenth terms of 

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{2,6,10,14,...)

We have

a10 - a4 = (10 - 4)d = 6(4) = 24

Arithmetic Series 

First we see that

1+ 2 + 3 + ... + 100 = 101 + 101 + ... + 101 (50 times) = 101(50)

In general

n(n + 1 + 2 + 3 + ... + n =2

Example 

What is

S = 1 + 4 + 7 + 10 + 13 +... + 46

Solution 

S = 1 + (1 + 1(3)) + (1 + 2(3)) + (1 + 3(3)) + ... + (1 + 15(3))

= (1 + 1 + ... + 1) + 3(1 + 2 + 3 + ... + 15)

= 16 + 3(15)(16)/2

In General

d(n - 1Sn = n (a1)+

2

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= 1/2 [2n(a1) + d(n - 1)(n)]

= 1/2[2n(a1)+ dn2 - dn]

= (n/2)[2(a1)+ dn - d]= (n/2)[2(a1) + d(n - 1)]

Or Alternatively

Sn = n/2(a1 + an)

 

Example 

How much will I receive over my 35 year career if my starting salary is $40,000, anreceive a 1,000 salary raise for each year I work here?

Solution

We have the series:

40,000 + 41,000 + 42,000 + ... + 74,000

= 35/2 (40,000 + 74,000) = $1,995,500

Convergent Series

Definition

Let {an} be a sequence, and letSn = a1 + a2 + ... + an, the nth partial sum.

If  Sn exists, we say that an is a convergent series, and write Sn = a

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Thus a series is convergent if and only if it's sequence of partial sums isconvergent. The limit of the sequence of partial sums is the sum of the series . series which is not convergent, is a divergent series .

Example The series r n is convergent with sum 1/(1 - r ), provided that | r | < 1. F

other values of r , the series is divergent; in particular, the series (- 1)n is divergen

Solution. We noted above that when | r | < 1, Sn a/(1 - r ) as n ; note particular cases;

= 1 or equivalently, + + + ... = 1.

Example 2 The sum is convergent with sum 1.

Solution. We can compute the partial sums explicitly:

Sn = = - = 1 - 1 as n .

Example 3 The sum is divergent.

Solution. We estimate the partial sums:

Sn =+ + + + + ... + + + ... + + ... +

>1 + + + > 2 if  n 15

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>1 + + + + > 3 if n 31

as n .

Example 4 The sum is convergent. [Actually the sum is /6, but this is muc

harder.]

Figure 1: Comparing the area under the curve y = 1/ x 2

with the area of the rectanglbelow the curve

Solution. We estimate the partial sums. Since 1/n2 > 0, clearly {Sn} is an increasingsequence. We show it is bounded above, whence by the Monotone ConvergenceTheorem , it is convergent. From the diagram,

+ + ... +<

, and so

Sn<1 + - 2 - .

Thus Sn < 2 for all n, the sequence of partial sums is bounded above, and the series convergent.

Proposition 1 Let an be convergent. Then an 0 as n .

Proof . Write l = Sn, and recall from our work on limits of sequences that Sn-1

as n . The

an = (a1 + a2 +...an) - (a1 + a2 +...an-1) = Sn - Sn-1 l - l  as n .

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Remark This gives a necessary condition for the convergence of a series; it is not  

sufficient. For example we have seen that 1/n is divergent, even though 1/n 0 an .

Example 5 The sum is divergent (Graphical method).

Solution. We estimate the partial sums. Since 1/n > 0, clearly {Sn} is an increasingsequence. We show it is not bounded above, whence by the note after , the sequencof partial sums as n .

Figure 2: Comparing the area under the curve y = 1/ x with the area of the rectangleabove the curve

From the diagram,

1 + + ... + > > + ... + .

Writing Sn = 1 + + ... + , we have Sn > log n > Sn - 1, or equivalently 1 + log n > Sn

log n for all n. Thus Sn and the series is divergent. [There is a much better 

estimate; the difference Sn - log n as n , where is Euler's constant.]

Proposition : Let an and bn be convergent. Then (an + bn) and c .an areconvergent.

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Proof . This can be checked easily directly from the definition; it is in effect thesame proof that the sum of two convergent sequences is convergent etc.

Divergent series

In mathematics, a divergent series is an infinite series that is not convergent,meaning that the infinite sequence of the partial sums of the series does nothave a limit.

If a series converges, the individual terms of the series must approach zero. Thus anseries in which the individual terms do not approach zero diverges. However,convergence is a stronger condition: not all series whose terms approach zeroconverge. The simplest counter example is the harmonic series

1+1/2+1/3+1/4+1/5+…….=1/n$ n!infinity

The divergence of the harmonic series was elegantly proven (here) by the medievalmathematician Nicole Oresme.

In specialized mathematical contexts, values can be usefully assigned to certain seriwhose sequence of partial sums diverges. A summability method or summationmethod is a partial function from the set of sequences of partial sums of series tovalues. For example, Cesàro summation assigns Grandi's divergent series

the value 1/2. Cesàro summation is an averaging method, in that it relies on the

arithmetic mean of the sequence of partial sums. Other methods involve analyticcontinuations of related series. In physics, there are a wide variety of summabilitymethods; these are discussed in greater detail in the article on regularization.

Theorems on methods for summing divergent series

 A summability method M is regular if it agrees with the actual limit on all convergentseries. Such a result is called an abelian theorem for M , from the prototypical Abel'stheorem. More interesting and in general more subtle are partial converse results,called tauberian theorems, from a prototype proved by Alfred Tauber . Here partial converse means that if M sums the series Σ, and some side-condition holds, then Σwas convergent in the first place; without any side condition such a result would say

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that M only summed convergent series (making it useless as a summation method fodivergent series).

The subject of divergent series, as a domain of mathematical analysis, is primarilyconcerned with explicit and natural techniques such as Abel summation, Cesàro

summation and Borel summation, and their relationships. The advent of Wiener'stauberian theorem marked an epoch in the subject, introducing unexpectedconnections to Banach algebra methods in Fourier analysis.

Summation of divergent series is also related to extrapolation methods and sequenctransformations as numerical techniques. Examples for such techniques are Padéapproximants, Levin-type sequence transformations, and order-dependent mappingsrelated to renormalization techniques for large-order perturbation theory in quantummechanics.

 

Regularity.

A summation method is regular  if, whenever the sequence s converges toA(s) =  x . Equivalently, the corresponding series-summation methevaluates AΣ(a) = x .

1. Linearity. A is linear  if it is a linear functional on the sequences where it

defined, so that A(r + s) = A(r ) + A(s) and A(ks) = k   A(s), for k a scalar (reacomplex.) Since the terms a n = s n+1 − s n of the series a are linear functionalsthe sequence s and vice versa, this is equivalent to A Σ being a linear functioon the terms of the series.

2.Stability. If s is a sequence starting from s 0 and s′ is the sequence obtainedomitting the first value and subtracting it from the rest, so that s′  n = s n+1 − s 0, thA(s) is defined if and only if  A(s′) is defined, and A(s) = s 0 + A(s′). Equivalenwhenever a′  n = a n+1 for all n, then A Σ(a) = a 0 + A Σ(a′).

The third condition is less important, and some significant methods, such as Bosummation, do not possess it.