MATHS PROJECT WORK

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MATHS PROJECT WORK
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SEQUENCE AND SERIES
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DEFINITION

• Sequence and Series, in mathematics, an ordered
  succession of numbers or other quantities, and
  the indicated sum of such a succession,
  respectively.

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IMPORTANT TYPES

• Important types of sequences include
  arithmetic sequences (also known as arithmetic
  progressions) in which the differences between
  successive terms are constant, and geometric
  sequences (also known as geometric progressions)
  in which the ratios of successive terms are
  constant.

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• The term series refers to the indicated
  sum, a1  a2  ...  an, or a1  a2  ...  an 
  ..., of the terms of a sequence. A series is
  either finite or infinite, depending on whether
  the corresponding sequence of terms is finite or
  infinite.
• The sequence s1  a1,s2  a1  a2, s3  a1  a2  
  a3, ..., sn  a1  a2  ...  an, ..., is called
  the sequence of partial sums of the
  series a1 a2  ...  an  .... The series
  converges or diverges as the sequence of partial
  sums converges or diverges.

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• A constant-term series is one in which the
  terms are numbers a series of functions is one
  in which the terms are functions of one or more
  variables. In particular, a power series is the
  series a0  a1(x - c)  a2(x - c)2  ...
   an(x - c)n  ..., in which c and the as are
  constants. In the case of power series, the
  problem is to describe what values of x they
  converge for. If a series converges for
  some x, then the set of all x for which it
  converges consists of a point or some connected
  interval.

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• The basic theory of convergence was worked out
  by the French mathematician Augustin Louis
  Cauchy in the 1820s.
• The theory and application of infinite series
  are important in virtually every branch of pure
  and applied mathematics.

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• Examples of sequences
• 1) 1, 2, 3, 4, 5, 6, 7, ...                add 1
  to the preceding term
• 2) 2, 4, 7, 11, 16, 23, 31.             add 2 to
  the preceding term, add 3 to the next term, etc
• 3) 1, 1, 2, 3, 5, 8, 13, 21, 34,...    add the
  two preceding terms together

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FIBONACCI SEQUENCE

• It was discovered by Leonardo of Pisa. This
  sequence occurs in nature, and Leonardo of Pisa
  derived it by studying the mating patterns of
  rabbits.
•  

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• Sequences and series can be finite or infinite.
  A finite sequence/series is one that eventually
  comes to an end, like the second one in the
  examples above. Infinite sequences/series are
  those that continue indefinitely, such as the
  first in the example as well as the Fibonacci
  sequence.
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• An arithmetic progression is a sequence in
  which each term (except the first term) is
  obtained from the previous term by adding a
  constant known as the common difference. An
  arithmetic series is formed by the addition of
  the terms in an arithmetic progression.
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• Let the first term on an A. P. be a and common
  difference d. Then,
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• General form of an A. P.
•             a, a  d, a  2d, a  3d, ...
•  nth term of an A. P.
•             a (n - 1) d
•  Sum of first n terms of an A. P.
•             n/2 2a (n - 1) d    or
•             n/2 first term last term

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• This idea was from the mathematician Carl
  Friedrich Gauss, who, as a young boy, stunned his
  teacher by adding up 1 2 3 ... 99 100
  within a few minutes. Here's how he did it
• He counted 101 terms in the series, of which 50
  is the middle term. He also realised that adding
  the first and last numbers, 1 and 100, gives,
  101 and adding the second and second last
  numbers, 2 and 99, gives 101, as well as 3 98
  101 and so on. Thus he concluded that there are
  50 sets of 101 and the middle term is 50.

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• sum of the series is
•                    50 (1 100) 50 5050.
• This can be rewritten as
•                    100/2 (1 100) 50 5050  
  or
•                    101/2 (1 100) 5050

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• Arithmetic mean.
• Given x, y and z are consecutive terms of an A.
  P., then
•                                  y - x  z - y
•                                     2y  x  z
•                                                   
            y is known as the arithmetic mean.
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GEOMETRIC PROGRESSION
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• A geometric progression is a sequence in which
  each term (except the first term) is derived from
  the preceding term by the multiplication of a
  non-zero constant, which is the common ratio. A
  geometric series is formed by the addition of the
  terms in a geometric progression.

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• Examples
• 1) 3, 6, 9, 12, ...                 first term 3,
  common ratio 3
• 2) 4, -8, 16, -32, ...           first term 4,
  common ratio -2

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•   
• Let the first term be a and common ratio be r.
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• General form of a G. P.
•               a, ar, ar2, ar3, ...
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• nth term of a G. P.
•                arn-1
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• Sum to first n terms of a G. P.
•               
•               
•  

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• Geometric mean. When x, y and z are
  consecutive numbers in a G. P.,
•                        y2  xz
•         y is the geometric mean

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CAUCHY SEQUENCE
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• In mathematics, a Cauchy sequence, named
  after Augustin Cauchy, is a sequence whose
  elements become arbitrarily close to each
  other as the sequence progresses. To be more
  precise, by dropping enough (but still only a
  finite number of) terms from the start of the
  sequence, it is possible to make the maximum of
  the distances from any of the remaining elements
  to any other such element smaller than any
  preassigned, necessarily positive, value.

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• In other words, suppose a pre-assigned
  positive real value  is chosen. However
  small  is, starting from a Cauchy sequence and
  eliminating terms one by one from the start,
  after a finite number of steps, any pair chosen
  from the remaining terms will be within
  distance  of each other.

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• The utility of Cauchy sequences lies in the
  fact that in a complete metric space (one where
  all such sequences are known to converge to a
  limit), they give a criterion for convergence
  which depends only on the terms of the sequence
  itself. This is often exploited in algorithms,
  both theoretical and applied, where an iterative
  process can be shown relatively easily to produce
  a Cauchy sequence, consisting of the iterates.

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• The notions above are not as unfamiliar as
  might at first appear. The customary acceptance
  of the fact that any real number x has a decimal
  expansion is an implicit acknowledgment that a
  particular Cauchy sequence of rational numbers
  (whose terms are the successive truncations of
  the decimal expansion of x) has the real limit x.
  In some cases it may be difficult to
  describe x independently of such a limiting
  process involving rational numbers.

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PRESENTED BY

• S.ANUPREETHI
• SANDHYA RANI PADHY
• SHUBHANGI SINGH
• KRITIKA MISHRA