Finite geometric series

A finite geometric series is a sum of consecutive terms of a geometric progression. With first term $a$ and $N$ terms, including $N=0$ for the empty sum,Unlike an infinite geometric series, this identity does not need $|r|<1$. For example, $3+6+12+24+48=3(2^5-1)=93$. The number of terms is five; the largest exponent is four.

The cancellation behind the formula is especially useful to remember: multiplying $S_N$ by the common ratio shifts all terms one place. In $S_N-r{S}_N$, every interior term cancels, leaving $a-a{r}^N$. Division by $1-r$ gives the identity when $r\ne 1$; at $r=1$ the sum is directly $Na$. The same argument works for complex numbers.

For integers $m\le n$ and $r\ne 1$, the shifted version isThere are $n-m+1$ terms. All powers must be defined; if a negative exponent is present, $r$ must be nonzero. At $r=1$ this sum is simply $n-m+1$. The finite trigonometric sum uses this identity with a complex exponential as the common ratio.

Finite sums of positive integers in a geometric progression already appear in Euclid's Elements, Book IX, Proposition 35. The argument there compares differences of proportional numbers; Joyce's guide rewrites its conclusion as the modern finite-sum formula. This historical result concerns finite sums, without asserting an infinite limit. ^[ref]

Differentiate the finite geometric series ${\sum }_{n=0}^N{r}^n=(1-r^{N+1})/(1-r)$ with respect to $r$, then multiply by $r$. This proves the title identity for $r\ne 1$ without any infinite-series differentiation. At $r=1$, the sum is $N(N+1)/2$. For $|r|<1$, both $N{r}^N$ and $N{r}^{N+1}$ tend to zero, yielding ${\sum }_{n=1}^{\infty }n{r}^n=r/(1-r{)}^2$.

Taking the real and imaginary parts of the finite geometric series ${\sum }_{m=1}^N{e}^{im\theta }$ yieldsThese formulas apply when $\theta \notin 2\pi \mathbb{Z}$. At multiples of $2\pi$, the sums are directly $N$ and $0$. Complex exponentials turn two real trigonometric summations into one finite geometric series.

For an interval $I$ of $L$ consecutive integers and $e(x)=e^{2\pi ix}$,where $\Vert \beta \Vert$ is the distance to the nearest integer. This follows from the finite geometric-series formula and $|1-e(\beta )|\asymp \Vert \beta \Vert$.