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 a
1, a
2, a
3…. a
n, and an infinite sequence is described by a
1, a
2, a
3…. to infinity. A sequence {a
n} 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 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 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 infinite.
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 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 QuestionsSUMMATIONS:
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:
.