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Mathematics | Sequence, Series and Summations

SEQUENCE:

It is a set of numbers in a definite order according to some definite rule (or rules).
Each number of the set is called a term of the sequence and its length is the number of terms in it. We can write the sequence as  {a_n}}_{n=1}^{infty} or a_n. A finite sequence is generally described by a1, a2, a3…. an, and an infinite sequence is described by a1, a2, a3…. to infinity. A sequence {an} has the limit L and we write displaystylelim_{n	oinfty} a_n = L or {a_n	oL} as {n	oinfty}.
For example:

2, 4, 6, 8 ...., 20 is a finite sequence obtained by adding 2 to the previous number.
10, 6, 2, -2, ..... is an infinite sequence obtained by subtracting 4 from the previous number. 

If the terms of a sequence can be described by a formula, then the sequence is called a progression.

1, 1, 2, 3, 5, 8, 13, ....., is a progression called the Fibonacci sequence in which each term 
is the sum of the previous two numbers.

More about progressions

Theorems:

Theorem 1: Given the sequence {a_n} if we have a function f(x) such that f(n) = a_n and displaystylelim_{x	oinfty} f(x) = L then displaystylelim_{n	oinfty} a_n = L. This theorem is basically telling us that we take the limits of sequences much like we take the limit of functions.

Theorem 2 (Squeeze Theorem): If a_nleq c_nleq b_n for all n > N for some N and lim_{n	oinfty} a_n = lim_{n	oinfty} b_n = L then lim_{n	oinfty} c_n = L.



Theorem 3: If lim_{n	oinfty}mid a_nmid = 0 then lim_{n	oinfty} a_n = 0 . Note that in order for this theorem to hold the limit MUST be zero and it won’t work for a sequence whose limit is not zero.

Theorem 4: If displaystylelim_{n	oinfty} a_n = L and the function f is continuous at L, then displaystylelim_{n	oinfty}f(a_n) = f(L)

Theorem 5: The sequence {r^n} is convergent if -1 < r leq 1 and divergent for
all other values of r. Also,


This theorem is a useful theorem giving the convergence/divergence and value (for when it’s convergent) of a sequence that arises on occasion.

Properties:

If (a_n) and (b_n) are convergent sequences, the following properties hold:

  • displaystylelim_{n	oinfty} (a_n pm b_n) = displaystylelim_{n	oinfty} a_n pm displaystylelim_{n	oinfty} b_n
  • displaystylelim_{n	oinfty} ca_n = cdisplaystylelim_{n	oinfty} a_n
  • displaystylelim_{n	oinfty} (a_n  b_n) = Big(displaystylelim_{n	oinfty} a_nBig)Big(displaystylelim_{n	oinfty} b_nBig)
  • displaystylelim_{n	oinfty} {a_n}^p = Big[displaystylelim_{n	oinfty} a_nBig]^p provided a_n geq 0
  • And the last property is

    SERIES:

    A series is simply the sum of the various terms of a sequence.
    If the sequence is a_1, a_2, a_3, ....a_n the the expression a_1 + a_2 + a_3 + ....+ a_n is called the series associated with it. A series is represented by ‘S’ or the Greek symbol displaystylesum_{n=1}^{n}a_n . The series can be finite or infinte.
    Examples:

    5 + 2 + (-1) + (-4) is a finite series obtained by subtracting 3 from the previous number.
    1 + 1 + 2 + 3 + 5 is an infinite series called the Fibonacci series obtained from the 
    Fibonacci sequence.
    

    If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite) then the series is also called convergent i.e. if displaystylelim_{n	oinfty} S_n = L then displaystylesum_{n=1}^infty} a_n = L. Likewise, if the sequence of partial sums is a divergent sequence (i.e. if displaystylelim_{n	oinfty} a_n 
eq 0 or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.

    Properties:

  • If displaystylesum_{n=1}^infty} a_n = A and displaystylesum_{n=1}^infty} b_n = B be convergent series then displaystylesum_{n=1}^infty} (a_n + b_n) = A + B
  • If displaystylesum_{n=1}^infty} a_n = A and displaystylesum_{n=1}^infty} b_n = B be convergent series then displaystylesum_{n=1}^infty} (a_n - b_n) = A - B
  • If displaystylesum_{n=1}^infty} a_n = A be convergent series then displaystylesum_{n=1}^infty} ca_n = cA
  • If displaystylesum_{n=1}^infty} a_n = A and displaystylesum_{n=1}^infty} b_n = B be convergent series then if a_nleq b_n for all n in N then Aleq B

    Theorems:

  • Theorem 1 (Comparison test): Suppose  0leq a_nleq b_n for ngeq k for some k. Then
    (1) The convergence of displaystylesum_{n=1}^infty} b_n implies the convergence of displaystylesum_{n=1}^infty} a_n.
    (2) The convergence of displaystylesum_{n=1}^infty} a_n implies the convergence of displaystylesum_{n=1}^infty} b_n.
  • Theorem 2 (Limit Comparison test): Let a_ngeq 0 and b_n > 0 , and suppose that displaystylelim_{n	oinfty}frac{a_n}{b_n} = L > 0. Then displaystylesum_{n=0}^infty} a_n converges if and only if displaystylesum_{n=0}^infty} b_n converges.
  • Theorem 3 (Ratio test): Suppose that the following limit exists, M = displaystylelim_{n	oinfty}frac{|a_n+1|}{|a_n|} . Then,
    (1) If M < 1 Rightarrow displaystylelim_{n	oinfty}a_n converges
    (2) If M > 1 Rightarrow displaystylelim_{n	oinfty}a_n diverges
    (3) If M = 1 Rightarrow displaystylelim_{n	oinfty}a_n might either converge or diverge
  • Theorem 4 (Root test): Suppose that the following limit exists:, M = displaystylelim_{n	oinfty}sqrt[n]{|a_n|} . Then,
    (1) If M < 1 Rightarrow displaystylelim_{n	oinfty}a_n converges
    (2) If M > 1 Rightarrow displaystylelim_{n	oinfty}a_n diverges
    (3) If M = 1 Rightarrow displaystylelim_{n	oinfty}a_n might either converge or diverge
  • Theorem 5 (Absolute Convergence test): A series displaystylesum_{n=1}^infty}a_n is said to be absolutely convergent if the series displaystylesum_{n=1}^infty}|a_n| converges.
  • Theorem 6 (Conditional Convergence test): A series displaystylesum_{n=1}^infty}a_n is said to be conditionally convergent if the series displaystylesum_{n=1}^infty}|a_n| diverges but the series displaystylesum_{n=1}^infty}a_n converges .
  • Theorem 7 (Alternating Series test): If a_0geq a_1geq a_2geq ....geq 0, and displaystylelim_{n	oinfty}a_n = 0, the the ‘alternating series’ a_0-a_1+a_2-a_3+.... = displaystylesum_{n=1}^infty}(-1)^n a_n will converge.
  • Series Questions

    SUMMATIONS:

    Summation is the addition of a sequence of numbers. It is is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.
    The summation symbol, displaystylesum_{i=m}^{n} a_i, instructs us to sum the elements of a sequence. A typical element of the sequence which is being summed appears to the right of the summation sign.

    Properties:

  • displaystylesum_{i=m}^{n} ca_i = cdisplaystylesum_{i=m}^{n}a_i where c is any number. So, we can factor constants out of a summation.
  • displaystylesum_{i=m}^{n} (a_ipm b_i) = displaystylesum_{i=m}^{n} a_i pmdisplaystylesum_{i=m}^{n} b_i So we can break up a summation across a sum or difference.
  • Note that while we can break up sums and differences as mentioned above, we can’t do the same thing for products and quotients. In other words,

  • displaystylesum_{i=m}^{n}a_i = displaystylesum_{i=m}^{j}a_i +displaystylesum_{i=j+1}^{n}a_i, for any natural number mleq j < j + 1leq p.
  • displaystylesum_{i=1}^{n}c = c+c+c+c....+(n times) = nc. If the argument of the summation is a constant, then the sum is the limit range value times the constant.
  • Examples:

    1) Sum of first n natural numbers: displaystylesum_{i=1}^{n}i = 1+2+3+....+n = frac{n(n+1)}{2}.
    
    2) Sum of squares of first n natural numbers: 
    displaystylesum_{i=1}^{n}i^2 = 1^2+2^2+3^2+....+n^2 = frac{n(n+1)(2n+1)}{6}.
    
    3) Sum of cubes of first n natural numbers: 
    displaystylesum_{i=1}^{n}i^3 = 1^3+2^3+3^3+....+n^3 = Bigg(frac{n(n+1)}{2}Bigg)^2.
    
    4) The property of logarithms: 
    displaystylesum_{i=1}^{n}log i = log 1+log 2+log 3+....+log n = log n!.
    


This article is attributed to GeeksforGeeks.org

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