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For $\mathrm{Re}s>1$, expansion of $(e^x-1{)}^{-1}$ as a geometric series and termwise integration give

For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

For $n\ge 3$ and a bounded smooth domain, a Dirichlet Laplacian eigenfunction with eigenvalue $\lambda$ satisfies the displayed estimate. A signed power test for a Laplacian eigenfunction and the zero-boundary Sobolev inequality give a Moser iteration with exponents $p_j=2[n/(n-2){]}^j$. The geometric series ${\sum }_{j}1/p_j=n/4$ gives the exact power of $\lambda$, while ${\sum }_j(\log p_j)/p_j<\infty$ controls the remaining constant. The constant depends only on dimension, because the zero-boundary Sobolev inequality follows by extension by zero to the whole space.

If $a$ is irrational and $b,c$ are real, the fractional parts of $a{n}^2+bn+c$ form an equidistributed sequence. In the Van der Corput lemma, every nonzero-lag correlation of $e^{2\pi im(a{n}^2+bn+c)}$ is a geometric series with irrational frequency $2mah$ and tends to zero. The Weyl criterion then applies.

A positional expansion in an integer base represents a rational number exactly when its digit sequence is eventually periodic. Long division proves the forward implication because only finitely many remainders occur; summing a periodic tail as a geometric series proves the converse.

A sinusoidal component at an integer multiple of a fundamental angular frequency. For a $2\pi$-periodic angular coordinate $\varphi$, real Fourier series use $\cos (k\varphi )$ and $\sin (k\varphi )$. Summing over $q$ equally spaced phases removes every component whose integer $k$ is not a multiple of $q$, since ${\sum }_{j=0}^{q-1}{e}^{2\pi ikj/q}=0$ in that case. Indeed, the geometric series has ratio $w=e^{2\pi ik/q}\ne 1$ and sum $(1-w^q)/(1-w)=0$. When $q$ divides $k$, all $q$ terms are one. This cancellation isolates higher-order feedback in resonant conjunction geometry.

For a primitive root of unity $\omega =e^{2\pi i/n}$ and labels $r,s$ modulo $n$, the displayed $n^2$ maximally entangled states form an orthonormal basis. Inner products equal ${\delta }_{s{s}^{\prime }}{n}^{-1}{\sum }_j{\omega }^{j(r^{\prime }-r)}={\delta }_{s{s}^{\prime }}{\delta }_{r{r}^{\prime }}$, by the geometric series. A nonprimitive root repeats phase labels and does not give a complete orthonormal basis.

For a geometric distribution count on the positive integers, independent of independent and identically distributed random variables $X_i$, the moment-generating function of $S={\sum }_{i=1}^N{X}_i$ is the displayed expression. Condition on the count and sum a geometric series. For real $t$, the finite domain requires $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. Positive summands always permit $t\le 0$, even when no positive exponential moment exists; this gives a Laplace transform of a nonnegative random variable. The aggregate has no mass at zero when all summands are strictly positive.

Starting Landweber iteration at zero with fixed step $\tau$, the full data-to-solution coefficient on a singular value $\sigma >0$ isThis follows by summing the scalar geometric series in a singular system of a compact operator. For fixed $n$, $g_n(\sigma )=n\tau \sigma +O({\sigma }^3)$ near zero and extends with $g_n(0)=0$. Its dimensionless damping factor is $\sigma g_n(\sigma )$, so the two common filter conventions must not be confused. Writing $\alpha =1/n$ uses a discrete parameter sequence; if negative multipliers occur, noninteger real powers are not a definition of the same iteration.