Geometric series

Table of contents
- Finite geometric series Geometric series
  - Finite trigonometric sum Finite geometric series
  - Exponential geometric sum bound Finite geometric series

Finite geometric series

For

r \neq = 1

, a finite geometric series satisfies

\sum_{j = m}^{n} r^{j} = r^{m} \frac{1 - r ^{n - m + 1}}{1 - r} .

Finite trigonometric sum

Taking the real and imaginary parts of the finite geometric series

\sum_{m = 1}^{N} e^{im θ}

yields

\sum_{m = 1}^{N} cos (m θ) = \frac{s i n ( Nθ /2 ) c o s (( N + 1 ) θ /2 )}{s i n ( θ /2 )}, \sum_{m = 1}^{N} sin (m θ) = \frac{s i n ( Nθ /2 ) s i n (( N + 1 ) θ /2 )}{s i n ( θ /2 )} .

These formulas apply when

θ \in / 2 π Z

. At multiples of

2 π

, the sums are directly

N

and

0

. Complex exponentials turn two real trigonometric summations into one finite geometric series.

Exponential geometric sum bound

For an interval

I

of

L

consecutive integers and

e (x) = e^{2 πi x}

,

\sum_{n \in I} e (β n) ≪ min (L, ∥ β ∥^{- 1}),

where

∥ β ∥

is the distance to the nearest integer. This follows from the finite geometric-series formula and

∣1 - e (β) ∣ ≍ ∥ β ∥

.

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Geometric series by

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A geometric series is a series of terms that have a constant ratio between successive terms. It is formed from a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

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