Mathematics | Sequence, Series and Summations

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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…. a_n, and an infinite sequence is described by a₁, a₂, a₃…. to infinity. A sequence {a_n} 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 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 $\displaystyle\lim_{n\to\infty} S_n = L$ 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: .

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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Last Updated : 25 Nov, 2022