Examples using Geometric Infinite Series Formula

Introduction to Infinite Geometric Series

An infinite geometric series is the sum of an infinite geometric sequence. This series would have no last term. The geometric series could also be described as geometric progression or the series that have successive terms having a common ratio. Hence, if the terms of geometric series approaches zero, the overall sum of its terms will be finite. The general form of a geometric sequence is: a, ar, ar2, ar3, ar4, ⋯. The nth term of a geometric sequence with initial value n and common ratio r is given by: a_n=arⁿ⁻¹.

In simple words, an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. Also known as a geometric series or progression.

Such a geometric sequence also follows the recursive relation:

a_n=ra_n−1

For every integer n≥1.

What Is Infinite Geometric Series Formula?

The general form of the infinite geometric series is a1+a1r+a1r²+a1r³+...a1+a1r+a1r²+a1r³+..., where a1a1 is the first term and r is the common ratio.

The sum to infinite GP means, the sum of terms in an infinite GP. The infinite geometric series formula is S∞ = a/ (1 – r), where a is the first term and r is the common ratio.

Sum of Infinite Geometric Series

Anyone can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if anyone add the larger numbers, individuals would not get a final answer. The only possible answer would be infinity. So, we don't deal with the common ratio greater than one for an infinite geometric series.

If the common ratio r lies between −1to 1, we can have the sum of an infinite geometric series. That is, the sum exits for | r |<1

The sum S of an infinite geometric series with −1<r<1is given by the formula,

S=a₁/1−r

An infinite series that has a sum is called a convergent series and the sum Sn is called the partial sum of the series.

Closed-form formula

As shown in the above information, the closed form of the geometric series partial sum up to and including the n-th power of r is a(1 - rⁿ⁺¹) / (1 - r) for any value of r, and the closed form of the geometric series is the full sum a / (1 - r) within the range |r| < 1.

Proof of convergence

The convergence of the geometric series depends on the value of the common ratio r:

1. If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r).
2. If |r| = 1, the series does not converge

Rate of convergence

Rate of Convergence is a measure of how fast the difference between the solution point and its estimates goes to zero. Faster algorithms usually use second-order information about the problem functions when calculating the search direction. A geometric series converges if the r- value (i.e. the number getting raised to a power) is between -1 and 1. A geometric series converges if and only if the absolute value of the common ratio, |r|, is less than 1.

Finding the Common Ratio of an Infinite Geometric Series

In geometric progression, the common ratio is the ratio between any one terms in the sequence and divide it by the previous term. Usually, it is represented by the letter “r”. Hence, anyone could calculate the common difference of a geometric progression by dividing any term by its prior term. Each and every term of the sequence, increases or decreases by a common factor or constant factor called the common ratio and eventually there will be an exponential growth towards positive infinity in the end.

Formula to calculate common ratio in geometric progression, a, ar, ar², ar³… is

Common ratio = a_n/a_n-1

For example, in simple words,

Common ratio = r = (2nd term)/ (first term) = a₂/a₁

Examples using Geometric Infinite Series Formula

For example, the sequence 2, 6, 18, 54, ⋯ 2, 6, 18, 54, ⋯ is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ⋯ 10, 5, 2.5, 1.25 , ⋯ is a geometric sequence with common ratio 12 . For every integer n≥1.

Another example: to find the sum of geometric series 125+ 25+ 5+ 1… following steps must be followed:

Solution:

The series is, 125 + 25 + 5 + 1 +…..

a₁ = 125

r = 25 /125 = 1/ 5

The formula for the resultant sum of the Infinite Geometric Series is,

S_∞ = a₁/ 1–r

S_∞ = 125/ 1– 1/5

S_∞ = 125/ 45

S_∞ = 625/ 4 

The following table shows several geometric series with different common ratios:

Therefore, a geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.

R= a₂ / a₁ = a₃/ a₂ =……..= a_n / a_n-1

The geometric sequence is sometimes called the geometric progression or GP, for short.