Mathematics | Sequence, Series and Summations

31 Mar, 2020
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 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  or  as .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
Theorem 1: Given the sequence  if we have a function f(x) such that f(n) =  and  then  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  for all n > N for some N and  then 
Theorem 3: If  then . 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  and the function f is continuous at L, then 
Theorem 5: The sequence  is convergent if  and divergent forall 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.
If  and  are convergent sequences, the following properties hold:
 provided 
And the last property is
A series is simply the sum of the various terms of a sequence.If the sequence is  the expression  is called the series associated with it. A series is represented by ‘S’ or the Greek symbol  . 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  then . Likewise, if the sequence of partial sums is a divergent sequence (i.e. if  or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.
If  and  be convergent series then 
If  and  be convergent series then 
If  be convergent series then 
If  and  be convergent series then if  for all n  N then Theorems:
Theorem 1 (Comparison test): Suppose  for  for some k. Then(1) The convergence of  implies the convergence of (2) The convergence of  implies the convergence of 
Theorem 2 (Limit Comparison test): Let  and , and suppose that . Then  converges if and only if  converges.
Theorem 3 (Ratio test): Suppose that the following limit exists, . Then,(1) If  converges(2) If  diverges(3) If  might either converge or diverge
Theorem 4 (Root test): Suppose that the following limit exists:, . Then,(1) If  converges(2) If  diverges(3) If  might either converge or diverge
Theorem 5 (Absolute Convergence test): A series  is said to be absolutely convergent if the series  converges.
Theorem 6 (Conditional Convergence test): A series  is said to be conditionally convergent if the series  diverges but the series  converges .
Theorem 7 (Alternating Series test): If , and , the ‘alternating series’  will converge.
Series Questions
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, , 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.
 where c is any number. So, we can factor constants out of a summation.
 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,
, for any natural number .
. 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: .

2) Sum of squares of first n natural numbers: 
.

3) Sum of cubes of first n natural numbers: 
.

4) The property of logarithms: 
.

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