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: