Sequence and series

A sequence is an ordered family indexed by the natural numbers; a series studies the partial sums of a sequence of terms.

A sequence is a function whose domain is usually the natural numbers.

A sequence is Fejér monotone with respect to a nonempty set $C$ if $\Vert z^{k+1}-p\Vert \le \Vert z^k-p\Vert$ for every $p\in C$. It is bounded. If it has a cluster point $\bar{z}\in C$, its nonincreasing distance to $\bar{z}$ and that convergent subsequence force the entire sequence to converge to $\bar{z}$.

A finite repetition-free sequence in a set $X$ is an injective function from an initial segment $\{0,\dots ,k-1\}$ of the natural numbers into $X$, including the empty sequence for $k=0$. For a finite set of size $m$, the number of these sequences isThe order of each list is part of the data, so it can be enumerated without choosing an order on its underlying support.

A real sequence $(a_n)$ is subadditive when $a_{m+n}\le a_m+a_n$ for all positive $m,n$. Fekete lemma givesallowing the value negative infinity.

If the real error depends only on the second index and ${\alpha }_n/n\to 0$, the normalized sequence has a limit in $[-\infty ,\infty )$. For fixed $k$, write $n=qk+r$ with $1\le r\le k$ and iterate to obtain $x_n\le x_r+q(x_k+{\alpha }_k)$. It follows that $\mathrm{limsup}{x}_n/n\le (x_k+{\alpha }_k)/k$ for every $k$; a subsequence attaining the lower limit proves convergence. The limit is ${\inf }_k(x_k+{\alpha }_k)/k$, even when the errors can be negative.

A geometric progression is a sequence whose successive terms have a constant ratio: $a_n=a{r}^n$ for constants $a$ and $r$.

A subsequence of $(a_n)$ has the form $(a_{n_k})$, where $n_1<n_2<\cdots$ is a strictly increasing sequence of indices.

Every real sequence has a monotone subsequence, without a boundedness assumption. Call an index a peak when its term is at least every later term. Infinitely many peaks give a nonincreasing subsequence; beyond finitely many peaks each term has a larger later term, which recursively gives an increasing subsequence.

A null sequence is a sequence that converges to the additive identity zero in its ambient topological vector space.

A sequence $(x_n)$ in a metric space is bounded when all its terms lie in some ball of finite radius.

A bounded subsequence is a subsequence whose terms all lie in one fixed bounded set.

A binary sequence is a finite or infinite sequence whose terms belong to $\{0,1\}$.

Partition the positive natural numbers into three infinite coordinate sets. Fix the coordinates on the first two sets to any prescribed binary values, and leave those on the third set free. Reading the third coordinate set embeds all infinite binary sequences in the resulting family, so the family is an uncountable set by Cantor's diagonal argument. For example, fixing $x_{3k-2}=0$ and $x_{3k-1}=a_{3k-1}$ while leaving $x_{3k}$ free guarantees infinitely many zeros and infinitely many agreements with any prescribed binary sequence $(a_n)$. Infinite free coordinates remain even after both requirements are enforced.

A real sequence is increasing when $a_{n+1}\ge a_n$ and decreasing when $a_{n+1}\le a_n$.

A real increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum. For the increasing case, the definition of supremum provides a term above $L-\epsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic monotone convergence theorem for integrals.

The limit superior of a real sequence $(a_n)$ isIt is the largest subsequential limit when the sequence is bounded.

The limit inferior of a real sequence $(a_n)$ isIt is the smallest subsequential limit when the sequence is bounded.

A series is the formal or limiting sum of the terms of a sequence.

For a series ${\sum }_n{a}_n$, let $r={\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}$. If $r<1$, choose $r<\rho <1$; eventually $|a_n|\le {\rho }^n$, so geometric comparison gives absolute convergence. If $r>1$, the terms fail to tend to zero along a subsequence and the series diverges. At $r=1$, the test is inconclusive.

A telescoping series has partial sums in which consecutive terms cancel, commonly because its summand has the form $b_n-b_{n+1}$.

If the terms $a_n$ of a series do not converge to zero, then ${\sum }_n{a}_n$ diverges. This is the contrapositive of $a_n=S_n-S_{n-1}\to 0$ for convergent partial sums.

An infinite product is the limit of the finite partial products ${\prod }_{n=1}^N{a}_n$. If $a_n=1+u_n$, every factor is nonzero, and ${\sum }_n|u_n|<\infty$, then the product converges to a nonzero value.

The Weierstrass elementary factor has a simple zero at $w=1$ and no other zeros. Its logarithmic series begins at degree $p+1$: $\log E_p(w)=-{\sum }_{\nu \ge p+1}{w}^{\nu }/\nu$ for $|w|<1$. Raising the factor order $p$ makes the tail arbitrarily small on a fixed compact set and enables infinite product convergence from logarithmic tails for unrestricted discrete zero data.

If a tail of holomorphic factors admits logarithms whose absolute values have a summable bound uniformly on every compact set, their product converges locally uniformly to the exponential of the logarithmic sum. The tail is holomorphic and nowhere zero. Finite initial factors may have zeros; those finite factors determine the locations and orders of all zeros near any given compact set. The scalar sufficient condition ${\sum }_j|u_j|<\infty$ for factors $1+u_j$ follows from $|\log (1+u_j)|\le 2|u_j|$ for $|u_j|\le 1/2$.

For $a_j\ge 0$, the finite product satisfiesThe lower inequality follows by product expansion; the upper one uses $1+t\le e^t$. Since both the partial sums and partial products are increasing, one converges to a finite limit exactly when the other does. The monotone bounded sequence theorem proves both implications. The product cannot tend to zero in this nonnegative setting.

This infinite product converges uniformly on compact subsets of the complex plane. Pair the factors at $iz$ and $-iz$ in the Weierstrass product for the reciprocal gamma function, and use the Gamma reflection formula together with $\Gamma (1+w)=w\Gamma (w)$. This gives $[\Gamma (1+iz)\Gamma (1-iz){]}^{-1}=\sinh (\pi z)/(\pi z)$ and the displayed product. The removable value at $z=0$ is one. Matching zeros alone would not exclude multiplication by a nonvanishing entire function; the normalized product fixes that ambiguity.

A lattice sum is a series indexed by the points of a lattice. For example, comparison with annular lattice-point counts shows that ${\sum }_{(m,n)\ne (0,0)}(m^2+n^2{)}^{-s}$ converges when $s>1$.

For terms $(a_n)$, the $n$th partial sum is $S_n={\sum }_{j=1}^n{a}_j$. A series converges when its sequence of partial sums converges.

Let $(b_n)$ be positive and increase to infinity. If the series ${\sum }_{n\ge 1}{x}_n/b_n$ converges, thenThis follows from summation by parts: write $x_k/b_k$ as a convergent series and average its tails with the increasing weights $b_k$.

The harmonic series diverges. Grouping terms from $2^k+1$ through $2^{k+1}$ gives at least $2^k/2^{k+1}=1/2$ from every block.

If $(a_n)$ is a nonnegative decreasing sequence, then ${\sum }_{n\ge 1}{a}_n$ converges if and only if the condensed series ${\sum }_{k\ge 0}{2}^k{a}_{2^k}$ converges. Comparing each dyadic block with its first and last terms proves both directions.

For a real subadditive sequence $(x_n{)}_{n\ge 1}$, the normalized sequence has the extended-real limit of a sequenceNo lower-bound hypothesis is needed for this conclusion. The limit of a sequence is finite precisely when the ratios are bounded below. The example $x_n=-n^2$ is a subadditive sequence with normalized limit of a sequence $-\infty$.

To prove the result, fix $k\ge 1$ and, for $n\ge k$, write $n=qk+r$ with $q\ge 1$ and $0\le r<k$. Iterating subadditivity gives $x_n\le q{x}_k+C_k$, where $C_k=\max (0,x_1,\dots ,x_{k-1})$ and $C_1=0$. Hence ${\mathrm{limsup}}_n{x}_n/n\le x_k/k$ for every $k$. Every ratio is at least their infimum. If that infimum is finite these bounds identify the limit of a sequence; if it is $-\infty$, choosing $k$ with an arbitrarily negative ratio gives convergence to $-\infty$.

Every bounded real sequence has a convergent subsequence. One proof repeatedly bisects a closed interval containing infinitely many terms, chooses a nested half containing infinitely many terms, and then chooses indices increasingly from those halves. Their interval diameters tend to zero, so completeness gives convergence.

A diagonal subsequence argument successively extracts nested subsequences that converge in the first, first two, and then first $k$ coordinates. Taking the $j$th term from the $j$th nested subsequence produces one subsequence that converges in every fixed coordinate.

If $I_1\supseteq I_2\supseteq \cdots$ are nonempty closed bounded intervals, then their intersection is nonempty. If their lengths tend to zero, the intersection consists of exactly one point.

If a bounded real sequence has the property that every convergent subsequence converges to the same number $L$, then the full sequence converges to $L$. Otherwise, a subsequence stays at least some fixed distance from $L$; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.

A real sequence $(x_n)$ has limit of a sequence $L\in \mathbb{R}$ when for every $\epsilon >0$ there is $N$ such that $|x_n-L|<\epsilon$ for all $n\ge N$. Its limit of a sequence is unique. Extended-real convergence to $-\infty$ means that every real upper bound eventually exceeds all the terms; convergence to $+\infty$ is the corresponding lower-bound condition. The extended-real Fekete lemma is an example where the limit of a sequence need not be finite.

The Cesaro mean of the first $n$ terms of a sequence is their arithmetic average. If $a_n\to L$, then its Cesaro means also tend to $L$.

If a sequence converges, its Cesaro means have the same limit. Given a small tolerance, split the average error into a fixed finite initial segment and a tail whose individual errors are below that tolerance. The first contribution is a fixed constant divided by $n$, while the second is bounded by the tolerance. The converse fails: the alternating sequence $(-1{)}^n$ has means tending to zero while its terms do not converge.

For a positive sequence converging to $L\ge 0$, its cumulative geometric means converge to the same limit.

An increasing real sequence converges exactly when it is bounded above. For a bounded sequence, its limit is the supremum of its set of terms.

The Fibonacci numbers satisfy $F_0=0$, $F_1=1$, and $F_{n+1}=F_n+F_{n-1}$.

For $\varphi =(1+\sqrt{5})/2$ and $\psi =(1-\sqrt{5})/2$, the Fibonacci numbers satisfy

For nonnegative $n,m,l$,Cassini's identity is the adjacent-index special case.

A pointwise limit assigns to each point the limit of the sequence of function values there. The convergence rate may depend on that point. Uniform convergence additionally bounds the error simultaneously over the whole domain.

A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

An exponential generating function encodes a sequence using the displayed factorial denominators. It is especially useful for counting labelled combinatorial structures: taking an unordered set of structures with positive size corresponds to exponentiating their exponential generating function. The rooted-tree generating function illustrates this through $T=z{e}^T$.

A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For ${\sum }_j{c}_j{z}^{m_j}$ with strictly increasing integer exponents, the radius of convergence satisfies $R^{-1}={\mathrm{limsup}}_j|c_j{|}^{1/m_j}$ by the Cauchy-Hadamard theorem. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.

For $|x|<1$, the binomial series is

For a nonnegative integer $L$ and $|z|<1$, this specialization of the binomial series sums the normalizing coefficients of a negative binomial distribution.

A power series $g(x)={\sum }_{\alpha }{b}_{\alpha }{x}^{\alpha }$ with nonnegative coefficients majorizes $f(x)={\sum }_{\alpha }{a}_{\alpha }{x}^{\alpha }$ when $|a_{\alpha }|\le b_{\alpha }$ for every multi-index $\alpha$. Coefficientwise majorization turns estimates for an analytic equation into estimates for a simpler positive-coefficient equation.

For a series ${\sum }_n{a}_n{x}^{m_n}$ with distinct increasing integer exponents, define the ordinary coefficients to be $a_n$ at $m_n$ and zero elsewhere. The Cauchy-Hadamard theorem then givesThe exponent in the root is the actual degree $m_n$, not the index $n$. For $a_n=(n!{)}^2$ and $m_n=n^2$, $0\le 2\log (n!)/n^2\le 2\log n/n\to 0$, so $R=1$. Boundary behavior is a separate question.

For a nonzero polynomial $p$, the power series ${\sum }_{n\ge 0}p(n)z^n$ has radius of convergence $1$. If $p$ has degree $d$ and leading coefficient $c\ne 0$, then $p(n)\sim c{n}^d$ and $|p(n+1)/p(n)|\to 1$, so the ratio test applies inside the unit disk. At every point of its boundary, the terms fail to tend to zero. The zero polynomial is the exceptional case with radius of convergence $\infty$.

Suppose real coefficients satisfy $c\le a_n\le C(1+n{)}^m$ for all sufficiently large $n$, where $c,C>0$ and $m\ge 0$. Then the power series $\sum a_n{z}^n$ has radius of convergence one. For $|z|<1$, the comparison test for series and the ratio test dominate it by a polynomial times a geometric sequence. For $|z|\ge 1$, the terms have modulus at least $c$ eventually and violate the term test for divergence. Finite changes to the coefficients do not affect the conclusion.

For $\sum a_n{z}^n$,with the usual conventions for zero and infinity.

If $L=\mathrm{limsup}{a}_n^{1/n}$ for positive coefficients, then the coefficients $a_n^2$ have root limsup $L^2$. Coefficients $a_n^{a_n}$ instead have root termsso their radius depends on the growth of $a_n\log a_n$ relative to $n$.

If the ordinary power series ${\sum }_{n\ge 0}{b}_n{w}^n$ has radius $R\in (0,\infty )$, thenconverges for $\mathrm{Re}z<\log R$ and diverges for $\mathrm{Re}z>\log R$. Behavior on the boundary depends on the coefficients.

The Cauchy product has coefficients $c_n={\sum }_{k=0}^n{a}_k{b}_{n-k}$. Absolute convergence permits regrouping and makes its sum the product of the two sums.

Inside its radius of convergence, a power series may be differentiated term by term, and the differentiated series has the same radius.

The first nonzero Taylor term controls a holomorphic function’s local magnitude and angular sign pattern.

A sequence of functions $f_n:X\to Y$ converges uniformly when one index $N$ makes $f_n(x)$ close to the limit simultaneously for every $x\in X$.

If continuously differentiable functions have uniformly convergent derivatives on an interval and their values converge at one point, the functions converge locally uniformly to a continuously differentiable limit whose derivative is the derivative limit. The proof integrates from the anchored point using the fundamental theorem of calculus. Uniform convergence of the functions on the entire interval follows when the interval is bounded; it need not follow on an unbounded interval.

On a finite-measure measurable set, a sequence of measurable functions converging pointwise almost everywhere converges uniformly outside a set of arbitrarily small measure. For tolerance $1/k$, the sets on which every index after $N$ has error at most $1/k$ increase to full measure as $N\to \infty$. Choose $N_k$ with exceptional measure at most $\epsilon 2^{-k-1}$ and intersect the good sets. Countable subadditivity bounds the discarded measure, and the resulting tail estimates prove uniform convergence. Finite measure is essential: $1_{[n,\infty )}$ on $[0,\infty )$ is a counterexample.

A uniform limit of continuous functions is continuous.

Dini's theorem says that a monotone sequence of continuous real-valued functions on a compact space that converges pointwise to a continuous function converges uniformly.

A sequence of functions is uniformly Cauchy when, for every $\epsilon >0$, all sufficiently late pairs satisfy

A sequence $f_n:X\to Y$ converges locally uniformly to $f$ when every point has a neighbourhood on which the convergence is uniform.

The compact-open topology on a space of functions is generated by uniform control on compact subsets. For scalar-valued functions, its basic seminorms are $f\mapsto {\sup }_{x\in K}|f(x)|$ as $K$ ranges over compact subsets of the domain.

A locally uniform limit of continuous real-valued functions is continuous. Moreover, local uniform convergence is uniform on every compact subset: choose finitely many neighbourhoods from the local uniformity cover and take the largest of their convergence indices.

For $r\ne 1$, a finite geometric series satisfies

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 $|1-e(\beta )|\asymp \Vert \beta \Vert$.

A sequence is Cauchy when its terms become arbitrarily close to one another beyond some index.

A real sequence converges if and only if the displayed condition holds. A Cauchy sequence makes all future differences uniformly small, so the forward implication is immediate. Conversely, failure of the Cauchy sequence criterion produces a fixed tolerance and, for each $n$, a future term differing from $a_n$ by at least half that tolerance. Choosing the least such positive lag defines one function that contradicts the displayed condition. The completeness of the real numbers then gives convergence. Testing only every fixed integer lag is weaker: $a_n=\log n$ passes those tests but diverges.

A Cauchy subsequence is a subsequence that is a Cauchy sequence. Every convergent subsequence is Cauchy.

Every Cauchy sequence of real numbers converges to a real number.

The least-upper-bound property states that every nonempty set of real numbers that has an upper bound has a supremum in the real numbers. It is an order-theoretic formulation of the completeness of the real numbers.

A series is conditionally convergent when it converges but does not converge absolutely.

If $a_n\ge 0$ decreases to zero, then ${\sum }_{n=1}^{\infty }(-1{)}^n{a}_n$ converges.

The alternating harmonic series ${\sum }_{n\ge 1}(-1{)}^{n+1}/n$ converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.

A rearrangement changes the order of a series without changing its multiset of terms. A conditionally convergent real series can be rearranged to approach any prescribed real limit or to diverge.

The series ${\sum }_{n=1}^{\infty }{n}^{-p}$ converges exactly when $p>1$.

A series converges absolutely when the series of absolute values converges; absolute convergence implies convergence.

If $a_n\ge 0$ and $\sum a_n<\infty$, then $a_n\to 0$, so eventually $a_n^2\le a_n$ and $\sum a_n^2$ converges.

The Cauchy-Schwarz inequality gives

If $a_n\ge 0$, $\sum a_n<\infty$, and $p>1/2$, thenThe endpoint can fail: $a_n=1/(n(\log n{)}^2)$ gives the divergent series $\sum 1/(n\log n)$.

The total variation of a sequence is ${\sum }_n|a_{n+1}-a_n|$; convergence alone does not make it finite.

Second-order total generalized variation combines a first derivative with an auxiliary vector field and its symmetric derivative. A standard continuous form isHere $Ew$ denotes the symmetric distributional derivative and the norms are total variations of the corresponding measures. Discrete variants replace these operators by linear maps and use sums of row norms. Unlike first-order total variation, this regularizer accommodates piecewise-affine behavior. A TGV divergence splitting makes its constrained divergence dual amenable to explicit proximal operators.

For a quadratic data term and divergence maps $A_1,A_2$, the dual constraint $A_{1}v\in D$ can be split through ${\delta }_D(A_{1}v)={\sup }_w[⟨w,A_{1}v⟩-{\delta }_D^{\ast }(w)]$. For a product of Euclidean row balls of radius $\beta$, its support function is ${\delta }_D^{\ast }(w)=\beta {\sum }_i\Vert w_i{\Vert }_2$. The resulting saddle coupling is $K(u,w)=A_2^{\ast }u-A_1^{\ast }w$. The Chambolle–Pock algorithm then uses a row-ball projection, a quadratic proximal operator and radial soft thresholding, with no projection onto an intersection involving a divergence operator.

Harmonic sums satisfy ${\sum }_{j=n+1}^{2n}1/j\to \log 2$ by integral comparison.

The notation $a_n\sim b_n$ means $a_n/b_n\to 1$.

Ratios of exponential functions are controlled by comparing their linear growth exponents.

Stirling’s formula gives n factorial asymptotic to square root of 2 pi n times (n/e)^n.

In Laplace method for the Gamma function, scaling $t=x(1+s/\sqrt{x})$ gives a standard Gaussian integral. The first even correction polynomial is $s^2-7s^4/12+s^6/18$. Its expectation under a unit-variance centered Gaussian is $1-7\cdot 3/12+15/18=1/12$. The amplitude expansion contributes to this coefficient as well as the phase expansion; expanding only the latter gives the wrong Stirling formula correction.

A series converges when its sequence of partial sums converges to a finite limit.