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 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.
Theorems:
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 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 and
are convergent sequences, the following properties hold:
provided
And the last property is
SERIES:
A series is simply the sum of the various terms of a sequence.
If the sequence isthe 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.
Properties:
- 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 ofimplies the convergence of
(2) The convergence ofimplies 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) Ifconverges
(2) Ifdiverges
(3) Ifmight either converge or diverge
- Theorem 4 (Root test): Suppose that the following limit exists:,
. Then,
(1) Ifconverges
(2) Ifdiverges
(3) Ifmight 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.
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,, 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:
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:
.