any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space,...

In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that...

for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp...

of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence...

rearranged or not without changing their sums using absolute convergence and conditional convergence of series. In modern terminology, any ordered infinite...

mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence...

(senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index) Each...

which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the...

approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic...

be rearranged to create arbitrary convergence. Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The general...

In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved...

if it is absolutely convergent. Absolute convergence – Mode of convergence of an infinite series Modes of convergence (annotated index) – Annotated index...

|a_{n}|} diverges is said to converge non-absolutely. It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an...

particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations...

indefinitely, but enclosing smaller areas each time. Absolute convergence Unconditional convergence Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill:...

of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. Monotone convergence theorem:...

M-test. The Cauchy convergence criterion states that a series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges if and only if the sequence...

index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships...

gives a sufficient condition for the convergence of expected values of random variables. Lebesgue's dominated convergence theorem. Let ( f n ) {\displaystyle...

economic growth literature the term "convergence" can have two meanings. The first kind (sometimes called "sigma-convergence") refers to a reduction in the...

continued fraction, converges absolutely. The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence. Finally, a continued...

which has a discontinuous pointwise limit. Another notion of convergence is uniform convergence. The uniform distance between two functions f , g : E → R...

region of absolute convergence is always a log-convex set in this sense.) On the other hand, in the interior of this region of convergence one may differentiate...

convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s0, then it automatically converges...

single point (in the case of absolute convergence), or the entire real number line (in the case of conditional convergence). In this formulation, Riemann's...

notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The...

only place where the absolute convergence is used). Since the series of the (an)n≥0 converges, the individual an must converge to 0 by the term test...

center of the disk of convergence, cn is the n-th complex coefficient, and z is a complex variable. The radius of convergence r is a nonnegative real...

radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur...

[citation needed] Assumption (3) is of more technical nature, implying absolute convergence and therefore allowing arbitrary rearrangement of an infinite series...