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 \displaystyle\lim_{n\to\infty} a_n = L or {a_n\to\L} as {n\to\infty}.
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 \displaystyle\lim_{x\to\infty} f(x) = L then \displaystyle\lim_{n\to\infty} 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_n\leq c_n\leq b_n for all n > N for some N and \lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L then \lim_{n\to\infty} c_n = L.



Theorem 3: If \lim_{n\to\infty}\mid a_n\mid = 0 then \lim_{n\to\infty} 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 \displaystyle\lim_{n\to\infty} a_n = L and the function f is continuous at L, then \displaystyle\lim_{n\to\infty}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:

  • \displaystyle\lim_{n\to\infty} (a_n \pm b_n) = \displaystyle\lim_{n\to\infty} a_n\ \pm \displaystyle\lim_{n\to\infty} b_n
  • \displaystyle\lim_{n\to\infty} ca_n = c\displaystyle\lim_{n\to\infty} a_n
  • \displaystyle\lim_{n\to\infty} (a_n  b_n) = \Big(\displaystyle\lim_{n\to\infty} a_n\Big)\Big(\displaystyle\lim_{n\to\infty} b_n\Big)
  • \displaystyle\lim_{n\to\infty} {a_n}^p = \Big[\displaystyle\lim_{n\to\infty} a_n\Big]^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 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 \displaystyle\sum_{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 \displaystyle\lim_{n\to\infty} S_n = L then \displaystyle\sum_{n=1}^\infty} a_n = L. Likewise, if the sequence of partial sums is a divergent sequence (i.e. if \displaystyle\lim_{n\to\infty} a_n \neq 0 or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.

    Properties:

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

    Theorems:

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

    SUMMATIONS:

    Summation is the addition of a sequence of numbers. It 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, \displaystyle\sum_{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:

  • \displaystyle\sum_{i=m}^{n} ca_i = c\displaystyle\sum_{i=m}^{n}a_i where c is any number. So, we can factor constants out of a summation.
  • \displaystyle\sum_{i=m}^{n} (a_i\pm b_i) = \displaystyle\sum_{i=m}^{n} a_i \pm\displaystyle\sum_{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,

  • \displaystyle\sum_{i=m}^{n}a_i = \displaystyle\sum_{i=m}^{j}a_i +\displaystyle\sum_{i=j+1}^{n}a_i, for any natural number m\leq j < j + 1\leq n.
  • \displaystyle\sum_{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: \displaystyle\sum_{i=1}^{n}i = 1+2+3+....+n = \frac{n(n+1)}{2}.
    
    2) Sum of squares of first n natural numbers: 
    \displaystyle\sum_{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: 
    \displaystyle\sum_{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: 
    \displaystyle\sum_{i=1}^{n}log\ i = log\ 1+log\ 2+log\ 3+....+log\ n = log\ n!.
    

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