** Arithmetic vs Geometric Series
**

The mathematical definition of a series is closely related to the sequences. A sequence is an ordered set of numbers and can be either a finite or an infinite set. A sequence of numbers with the difference between two elements being a constant is known as an arithmetic progression. A sequence with a constant quotient of two successive numbers is known as a geometric progression. These progressions can either be finite or infinite, and if finite, number of terms is countable, else uncountable.

Generally, the sum of the elements in a progression can be defined as a series. The sum of an arithmetic progression is known as an arithmetic series. Likewise, the sum of a geometric progression is known as a geometric series.

**More about Arithmetic Series**

In an arithmetic series, the successive terms have a constant difference.

S_{n }= a_{1 }+ a_{2 }+ a_{3 }+ a_{4 }+⋯+ a_{n }= ∑^{n}_{i=1 }a_{i} ; where a_{2 }= a_{1 }+ d, a_{3 }= a_{2 }+ d, and so on.

This difference d is known as the common difference, and the n^{th} term is given by a_{n }= a_{1}+ (n-1)d; where a_{1} is the first term.

The behaviour of the series changes based on the common difference d. If the common difference is positive the progression tends to be positive infinity, and if the common difference is negative it tends towards the negative infinity.

The sum of the series can be obtained by the following simple formula, which was first developed by Indian astronomer and mathematician Aryabhata.

S_{n }= n/2 (a_{1}+ a_{n} ) = n/2 [2a_{1 }+ (n-1)d]

The sum S_{n} can either be finite or infinite, based on the number of terms.

**More about Geometric Series**

A geometric series is a series with the quotient of the successive numbers constant. It is an Important series found in the study of the series, because of the properties it possesses.

S_{n }= ar + ar^{2 }+ ar^{3 }+⋯+ ar^{n }= ∑^{n}_{i=1 }ar^{i}

Based on the ratio r, the behaviour of the series can be categorized as follows. r={|r|≥1 series diverges; r≤1 series converges}. Also, if r<0 the series oscillates, i.e. the series has alternating values.

The sum of the geometric series can be calculated using the following formula. S_{n }= a(1-r^{n}) / (1-r); where a is the initial term and r is the ratio. If the ratio r≤1, the series converges . For an infinite series, the value of convergence is given by S_{n}= a / ( 1-r).

Geometric series has numerous applications in the fields of physical sciences, engineering, and economics

**What is the difference between Arithmetic and Geometric Series?**

• An arithmetic series is a series with a constant difference between two adjacent terms.

• A geometric series is a series with a constant quotient between two successive terms.

• All infinite arithmetic series are always divergent, but depending on the ratio, the geometric series can either be convergent or divergent.

• The geometric series can have oscillation in the values; that is, the numbers change their signs alternatively, but the arithmetic series cannot have oscillations.

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