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← Sums of Geometric Series

Rule

Sum of a Geometric Series

Let a_1 and r be the first term and the common ratio, respectively, of a geometric sequence with n terms, where r≠1. The sum of the related finite geometric series or the partial sum of the infinite series can be found by using the following formula.

S_n=a_1(1-r^n)/1-r, r≠ 1

For an infinite series, if the common ratio r is greater than - 1 and less than 1 — in other words, if |r|<1 — then the sum can be found by using the following formula.

S_(∞)=a_1/1-r, - 1

This means that the sum converges on a number. If the common ratio r is less than or equal to - 1 or greater than or equal to 1 — if |r| ≥ 1 — then the sum diverges. In such cases, there is no sum for the infinite geometric series.

Proof

Finite Sum

To derive the formula for a geometric series, a geometric sequence of n terms with the first term a_1 and the common ratio r will be considered. a_1, a_1r, a_1r^2, ... , a_1r^(n-1) All the terms will be added to express the series and a_1 will be factored out.

S_n=a_1+a_1r+a_1r^2+...+a_1r^(n-1)

Factor out a_1

S_n=a_1(1+r+r^2+...+r^(n-1))

Then, both sides of the equation will be multiplied by (1-r) and the resulting equation simplified.

S_n=a_1(1+r+r^2+...+r^(n-1))

LHS * (1-r)=RHS* (1-r)

S_n(1-r)=a_1(1+r+r^2+...+r^(n-1))(1-r)

Distribute (1-r)

S_n(1-r)=a_1(1(1-r)+r(1-r)+r^2(1-r)+...+r^(n-1)(1-r))

Multiply

S_n(1-r)=a_1(1-r+r-r^2+r^2-r^3+...+r^(n-1)-r^n)

Notice that the like terms are ordered in minus-plus pairs. This means that after simplifying, they will cancel out and only the first and last terms will remain.

S_n(1-r)=a_1(1-r+r-r^2+r^2-r^3+...+r^(n-2)-r^(n-1)+r^(n-1)-r^n)

Add and subtract terms

S_n(1-r)=a_1(1-r^n)

.LHS /(1-r).=.RHS /(1-r).

S_n=a_1(1-r^n)/1-r

The formula for the sum of a finite geometric series has been derived.

S_n=a_1(1-r^n)/1-r

Proof

Infinite Sum

To derive the formula for the sum of an infinite geometric series with -1number between -1 and 1, the value of r^n becomes very small as the value of n increases. In other words, it gets closer to 0 as n approaches infinity. r^n [n→ ∞] 0 Therefore, r^n=0 can be substituted into the standard formula and the resulting equation simplified.

S_n=a_1(1-r^n)/1-r

r^n= 0, n= ∞

S_(∞)=a_1(1- 0)/1-r

Subtract term

S_(∞)=a_1 (1)/1-r

Identity Property of Multiplication

S_(∞)=a_1/1-r

The formula for the sum of an infinite geometric series with -1

S_(∞)=a_1/1-r

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