How To Find If A Sequence Is Convergent Or Divergent

Number Sequence Computer

Arithmetic Sequence Reckoner

definition: a_n = a_one + f × (n-1)
case: 1, 3, 5, 7, nine 11, 13, ...

Geometric Sequence Calculator

definition: a_n = a × r_n-1
example: 1, 2, 4, 8, 16, 32, 64, 128, ...

Fibonacci Sequence Calculator

definition: a₀=0; a_i=1; a_north = a_n-1 + a_n-2;
example: 0, 1, one, two, 3, v, 8, 13, 21, 34, 55, ...

In mathematics, a sequence is an ordered list of objects. Accordingly, a number sequence is an ordered list of numbers that follow a particular design. The private elements in a sequence is often referred to as term, and the number of terms in a sequence is called its length, which can be infinite. In a number sequence, the order of the sequence is important, and depending on the sequence, it is possible for the same terms to announced multiple times. There are many dissimilar types of number sequences, three of the nigh common of which include arithmetic sequences, geometric sequences, and Fibonacci sequences.

Sequences have many applications in diverse mathematical disciplines due to their properties of convergence. A series is convergent if the sequence converges to some limit, while a sequence that does not converge is divergent. Sequences are used to written report functions, spaces, and other mathematical structures. They are peculiarly useful as a ground for serial (essentially describe an operation of adding infinite quantities to a starting quantity), which are generally used in differential equations and the surface area of mathematics referred to equally analysis. At that place are multiple ways to denote sequences, one of which involves but listing the sequence in cases where the pattern of the sequence is hands discernible. In cases that have more complex patterns, indexing is unremarkably the preferred notation. Indexing involves writing a full general formula that allows the determination of the n^thursday term of a sequence every bit a office of n.

Arithmetic Sequence

An arithmetic sequence is a number sequence in which the difference between each successive term remains constant. This difference can either be positive or negative, and dependent on the sign will result in terms of the arithmetic sequence tending towards positive or negative infinity. The general form of an arithmetic sequence can be written as:

	a_n = a₁ + f × (due north-ane) or more by and large	where a_n refers to the n^th term in the sequence
	a_northward = a_k + f × (n-m)	a₁ is the first term
i.eastward.	a_i, a_i + f, a_one + 2f, ...	f is the common divergence
EX:	1, 3, v, vii, 9, eleven, thirteen, ...

Information technology is clear in the sequence higher up that the mutual difference f, is 2. Using the equation to a higher place to calculate the 5^th term:

EX:

a_v = a₁ + f × (n-ane)
a₅ = 1 + 2 × (5-one)
a₅ = 1 + 8 = nine

Looking back at the listed sequence, information technology can be seen that the 5th term, a₅ , constitute using the equation, matches the listed sequence as expected. Information technology is as well commonly desirable, and simple, to compute the sum of an arithmetic sequence using the following formula in combination with the previous formula to notice a_northward :

Using the same number sequence in the previous instance, notice the sum of the arithmetic sequence through the 5^thursday term:

EX:

1 + 3 + v + 7 + 9 = 25
(5 × (1 + 9))/two = 50/2 = 25

Geometric Sequence

A geometric sequence is a number sequence in which each successive number after the first number is the multiplication of the previous number with a stock-still, non-nil number (common ratio). The general form of a geometric sequence tin be written as:

	a_n = a × r^n-one	where a_n refers to the n^th term in the sequence
i.east.	a, ar, ar², ar³, ...	a is the calibration factor and r is the mutual ratio
EX:	one, 2, four, 8, 16, 32, 64, 128, ...

In the example above, the common ratio r is two, and the calibration cistron a is 1. Using the equation in a higher place, calculate the eight^thursday term:

EX:

a₈ = a × r^8-i
a₈ = 1 × 2⁷ = 128

Comparing the value institute using the equation to the geometric sequence above confirms that they friction match. The equation for computing the sum of a geometric sequence:

Using the same geometric sequence above, discover the sum of the geometric sequence through the three^rd term.

EX: 1 + ii + iv = 7

Fibonacci Sequence

A Fibonacci sequence is a sequence in which every number following the kickoff two is the sum of the two preceding numbers. The first ii numbers in a Fibonacci sequence are defined as either i and 1, or 0 and i depending on the chosen starting point. Fibonacci numbers occur often, every bit well equally unexpectedly within mathematics and are the subject of many studies. They have applications within calculator algorithms (such as Euclid'south algorithm to compute the greatest common factor), economics, and biological settings including the branching in copse, the flowering of an artichoke, also every bit many others. Mathematically, the Fibonacci sequence is written as:

	a_n = a_n-one + a_northward-2	where a_{due north} refers to the due north^th term in the sequence
EX:	0, one, ane, 2, 3, 5, 8, 13, 21, ...	a₀ = 0; a₁ = 1