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Geometric series — convergence and sum

Calculate the sum of the geometric series Σ_{n=0}^{∞} (3/4)^n.

Given data

Σ_{n=0}^{∞} (3/4)^n

Steps

0 / 3

Identify the ratio q of the geometric series.
Write your working — the numerical result must appear in the text (adim.)
Does the series converge for |q| < 1? Verify.
Apply the sum formula: S = 1/(1−q).

Full worked solution

Identify the ratio q of the geometric series.
$q = \dfrac{3}{4} = 0.75$
The series is $\sum_{n=0}^\infty q^n$ with $q = \frac{3}{4} = \mathbf{0.75}$ . The ratio is the constant factor between consecutive terms.
Does the series converge for |q| < 1? Verify.
$|0.75| = 0.75 < 1 \Rightarrow \text{converge}$
$|q| = 0.75 < 1$ → the series converges. A geometric series converges absolutely when the absolute value of the ratio is less than 1.
Apply the sum formula: S = 1/(1−q).
$S = \dfrac{1}{1 - 0.75} = \dfrac{1}{0.25}$
$S = \frac{1}{1-q} = \frac{1}{1-0.75} = \frac{1}{0.25} = \mathbf{4}$ . The sum of the convergent geometric series is $4$ .

Result:The series converges to S = 4.

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