Geometric Series · aiwiki.page

A geometric series is a sum whose successive terms are obtained by multiplying the preceding term by a fixed number, called the common ratio. With initial term $a$ and common ratio $r$ , it has the form

a+ar+ar^2+ar^3+\cdots.

A finite geometric series contains a specified number of terms; an infinite series is interpreted through the limit of its partial sums. For $a\ne0$ , the infinite geometric series converges precisely when $|r|<1$ , and its sum is $a/(1-r)$ . (openstax.org)

Definition and notation

The terms of a geometric series form a geometric sequence. The sequence lists the terms, whereas the series adds them. For example, $3,6,12,24$ is a geometric sequence with common ratio $2$ , and $3+6+12+24$ is its corresponding finite series. (openstax.org)

Using summation notation, the sum of the first $N$ terms is

S_N=\sum_{n=0}^{N-1}ar^n.

Thus the final term is $ar^{N-1}$ , not $ar^N$ . The same expression may be indexed from $1$ :

S_N=\sum_{n=1}^{N}ar^{n-1}.

Changing the indexing does not change the series, provided the exponents and limits are adjusted consistently. (openstax.org)

The initial term and ratio may be real numbers or complex numbers. Defining terms through multiplication, rather than through quotients of consecutive terms, also accommodates $r=0$ : the resulting series is $a+0+0+\cdots$ . The term with exponent zero is understood as the initial term $a$ . (en.wikipedia.org)

Finite sums and their derivation

For $r\ne1$ ,

\boxed{S_N=a\frac{1-r^N}{1-r}}.

The formula follows from a short algebraic calculation. Multiplying the sum by $r$ gives

rS_N=ar+ar^2+\cdots+ar^N.

Subtracting this from the original expression cancels all intermediate terms:

(1-r)S_N=a-ar^N.

Division by $1-r$ yields the formula. If $r=1$ , every term equals $a$ , so instead $S_N=Na$ . (openstax.org)

For example,

3+6+12+24 =3\frac{1-2^4}{1-2} =45.

The finite-sum formula applies whether the terms grow or shrink; no condition such as $|r|<1$ is needed for a finite sum. (openstax.org)

Infinite sums and convergence

An infinite series has sum $S$ when its partial sums have the limit

S=\lim_{N\to\infty}S_N.

If $|r|<1$ , then $r^N\to0$ , and the finite-sum formula gives

\boxed{\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}}.

For example,

1+\frac12+\frac14+\frac18+\cdots=2, \qquad 1-\frac12+\frac14-\frac18+\cdots=\frac23.

These equalities describe limits of finite sums, not the completion of an infinite sequence of additions. (openstax.org)

For $a\ne0$ , the cases outside the convergence condition are:

$r=1$ : the partial sums are $Na$ and do not approach a finite limit.
$r=-1$ : the partial sums alternate between $a$ and $0$ .
$|r|>1$ : the terms do not tend to zero.
Complex $r$ with $|r|=1$ : the terms have constant nonzero magnitude and therefore cannot tend to zero.

A necessary condition for convergence of any series is that its terms tend to zero. This establishes divergence in the latter cases. The exceptional case $a=0$ gives the zero series for every ratio. (en.wikipedia.org)

Every convergent geometric series is also absolutely convergent, because

\sum_{n=0}^{\infty}|ar^n| =|a|\sum_{n=0}^{\infty}|r|^n =\frac{|a|}{1-|r|}.

Consequently, geometric series provide standard comparison series in mathematical analysis. Comparisons with geometric decay underlie the ratio test and root test for more general series. (openstax.org)

Remainders and approximation

Subtracting the finite sum from the infinite sum gives the exact remainder after $N$ terms:

R_N=S-S_N=\frac{ar^N}{1-r}, \qquad |r|<1.

Its magnitude is

|R_N|=\frac{|a|\,|r|^N}{|1-r|} \leq\frac{|a|\,|r|^N}{1-|r|}.

These expressions follow directly from the two sum formulas and quantify the error made by truncating the series. (openstax.org)

For a fixed ratio, increasing $N$ multiplies the remainder’s magnitude by $|r|$ at each step. Ratios whose magnitude is close to $1$ therefore give slower convergence than ratios of smaller magnitude. For real $a>0$ , positive ratios below $1$ produce partial sums that increase toward the sum; negative ratios above $-1$ produce partial sums that alternate around it. (openstax.org)

Power series and calculus

Allowing the ratio to vary gives the fundamental power series identity

\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n, \qquad |x|<1.

Its radius of convergence is $1$ ; neither real endpoint $x=1$ nor $x=-1$ is included. Substitution generates expansions for other functions, such as

\frac{1}{1+x^2} =\sum_{n=0}^{\infty}(-1)^n x^{2n}, \qquad |x|<1.

(openstax.org)

Within the interval of convergence, power series may be differentiated and integrated term by term. Taking the derivative of the geometric expansion produces

\frac{1}{(1-x)^2} =\sum_{n=1}^{\infty}n x^{n-1}.

Taking an integral from $0$ to $x$ produces

-\log(1-x)=\sum_{n=1}^{\infty}\frac{x^n}{n}, \qquad |x|<1.

These derived series are not themselves geometric: their coefficients vary with $n$ . (openstax.org)

Repeating decimals

Geometric series express repeating decimals as rational numbers. For example,

0.\overline{27} =\frac{27}{100}+\frac{27}{100^2}+\frac{27}{100^3}+\cdots =\frac{27/100}{1-1/100} =\frac{3}{11}.

More generally, a repeating block of $k$ digits, interpreted as the integer $B$ and beginning immediately after the decimal point, represents

0.\overline{\text{block}}=\frac{B}{10^k-1}.

This is an application of the infinite-sum formula with common ratio $10^{-k}$ . (openstax.org)

Historical geometric interpretation

Archimedes used an argument corresponding to a geometric series in his quadrature of a parabolic segment. Starting with an inscribed triangle of area $A$ , successive collections of additional triangles contributed areas

\frac A4,\quad \frac A{16},\quad \frac A{64},\ldots.

In modern notation, the total is

A\sum_{n=0}^{\infty}\left(\frac14\right)^n=\frac{4A}{3}.

His proof used the method of exhaustion, rather than modern notation for infinite series, to establish the area of the segment. It provides an early geometric instance of summing quantities that decrease by a fixed proportion. (arxiv.org)

References

2 Infinite Series - Calculus Volume 2openstax.org
4 Series and Their Notations - Precalculus 2eopenstax.org
6 Ratio and Root Tests - Calculus Volume 2openstax.org
Ch. 5 Key Equations - Calculus Volume 2openstax.org
1 Power Series and Functions - Calculus Volume 2openstax.org
Ch. 6 Key Concepts - Calculus Volume 2openstax.org
Geometric seriesen.wikipedia.org
Archimedes' quadrature of the parabola and minimal coversarxiv.org