Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is obtained by multiplying the previous term by the same constant number called the common ratio.

1 + 2 + 4 + 8 + 16 + . . .

Here, each term is multiplied by 2, so the common ratio is r = 2.

The general form of a geometric series is:

a + ar + ar² + ar³ + ⋯

• a = first term
• r = common ratio

Below is the visual representation of Geometric series

The large square represents a whole area of 1. It is repeatedly divided into smaller squares whose areas form the sequence:

\frac{1}{4}, \ \frac{1}{8}, \ \frac{1}{16},\ \frac{1}{32},\ \frac{1}{64}, \ \cdots\

So the shaded regions form the geometric series

\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\cdots

Convergence of Geometric Series

The convergence of a geometric series (infinite) depends solely on the value of the common ratio r:

• Convergent Series: The series converges if the absolute value of the common ratio is less than 1: ∣r∣ < 1
• Divergent Series: The series diverges if the absolute value of the common ratio is equal to or greater than 1: ∣r∣ ≥ 1

Formula

The Geometric Series formula for the Finite series is given as,

\bold{{S_n =\frac{a(1-r^n)}{1-r}}}

Where

• S_n = sum up to n^th term,
• a = First term, and
• r = common factor.

Derivation for Geometric Series Formula

Suppose a Geometric Series for n terms: 

S_n = a + ar + ar² + ar³ + .... + ar^n-1 . . . (1)

Multiplying both sides by the common factor (r):

r S_n = ar + ar² + ar³ + ar⁴ + ... + arⁿ . . . (2)

Subtracting Equation (1) from Equation (2):

(r S_n - S_n) = (ar + ar² + ar³ + ar⁴ +. . . arⁿ) - (a + ar + ar² + ar³ + . . . + ar^n-1)

⇒ S_n (r-1) = arⁿ - a

⇒ S_n (1 - r) = a (1-rⁿ)

⇒ {S_n =\frac{a(1-r^n)}{1-r}}

Note: When the value of k starts from 'm', the formula will change.

\sum_{k=m}^{n}ar^k=\frac{a(r^m-r^{n+1}}{1-r}, when r≠0

For Infinite Geometric Series

n will tend to Infinity, n ⇢ ∞, Putting this in the generalized formula:

S_\infty = \sum_{n=1}^{\infty}ar^{n-1} = \frac{a}{1-r}; -1<{r}<1

n^th term for the G.P. : a_n = ar^n-1

Geometric Sequence vs Geometric Series

Some of the common differences between Geometric Sequences and Series are listed in the following table:

Related Articles

• Sequences and Series in Maths
• Sequences and Series Formulas
• Arithmetic Progression
• Geometric Progression
• Practice Problems on Geometric Series