Series

A series is the sum of the terms of a sequence. Series let us add up infinitely many contributions in closed form — the trick behind expected values of count distributions and many probability generating functions.

Partial sums#

Given a sequence $\{a_n\}$ , the $N$ -th partial sum is $S_N = \sum_{n=1}^{N} a_n$ . An infinite series $\sum_{n=1}^{\infty} a_n$ converges to $S$ if $S_N \to S$ as $N \to \infty$ ; otherwise it diverges.

Arithmetic series#

With first term $a$ and common difference $d$ , the terms are $a, a+d, a+2d, \dots$ and the finite sum is

$\sum_{i=1}^{n} \big(a + (i-1)d\big) = \frac{n}{2}\,\big(2a + (n-1)d\big).$

Geometric series#

With ratio $r$ , the terms are $a, ar, ar^2, \dots$ . When $|r| < 1$ the infinite series converges:

$\sum_{n=1}^{\infty} a\,r^{\,n-1} = \frac{a}{1 - r}.$

Show why the sum is

a/(1-r)

Write the partial sum of the first $N$ terms and multiply the whole thing by $r$ :

$S_N = a + ar + ar^2 + \cdots + ar^{N-1}, \qquad rS_N = ar + ar^2 + \cdots + ar^{N}.$

Subtracting cancels every interior term (a telescoping trick):

$S_N - rS_N = a - ar^{N} \;\Longrightarrow\; S_N = \frac{a\,(1 - r^{N})}{1 - r}.$

When $|r| < 1$ the term $r^{N} \to 0$ as $N \to \infty$ , so the partial sums converge:

$\sum_{n=1}^{\infty} a\,r^{\,n-1} = \frac{a}{1 - r}.$

If $|r| \ge 1$ the term $r^{N}$ does not vanish and the series diverges.

Power series#

A power series $\sum_{n=0}^{\infty} c_n x^n$ defines a function on its interval of convergence. The key example is

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

Note the harmonic series $\sum_{n=1}^{\infty} \tfrac{1}{n}$ diverges, even though its terms shrink to $0$ — small terms are not enough for convergence.

Worked example#

Let $a = 1$ , $r = 1/2$ . The infinite geometric series is

$\sum_{n=1}^{\infty} \left(\tfrac{1}{2}\right)^{n-1} = \frac{1}{1 - \tfrac12} = 2.$

The partial sum to $n = 10$ terms is $2 - (1/2)^{9} \approx 1.99805$ , already very close to $2$ .

Computing it#

R#

r <- 0.5; a <- 1; n <- 10
partial <- sum(a * r^(0:(n - 1)))   # 1.998047
closed  <- a / (1 - r)              # 2
c(partial, closed)

Python#

Python

r, a, n = 0.5, 1, 10
partial = sum(a * r**k for k in range(n))   # 1.998046875
closed  = a / (1 - r)                        # 2.0
print(partial, closed)

1.998046875 2.0

Julia#

Julia

r, a, n = 0.5, 1, 10
partial = sum(a * r^k for k in 0:(n - 1))   # 1.998046875
closed  = a / (1 - r)                        # 2.0
partial, closed

Why it matters for statistics#

Geometric series give the mean of the geometric distribution and normalize infinite discrete distributions. Power series underlie moment and probability generating functions, and Taylor series (a special power series) drive approximations like the delta method. Recognizing when a series converges tells you whether an expectation is even finite.