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Wynn's epsilon method

Wynn's -method is a method for numerical evaluation of sums and products that samples a number of additional terms in the series and then tries to extrapolate them by fitting them to a polynomial multiplied by a decaying exponential.In particular, the method provides an efficient algorithm for implementing transformations of the form(1)where(2)is the th partial sum of a sequence , which are useful for yielding series convergence improvement (Hamming 1986, p. 205). In particular, letting , , and(3)for , 2, ... (correcting the typo of Hamming 1986, p. 206). The values of are there equivalent to the results of applying transformations to the sequence (Hamming 1986, p. 206).Wynn's epsilon method can be applied to the terms of a series using the Wolfram Language command SequenceLimit[l]. Wynn's method may also be invoked in numerical summation and multiplication using Method -> Fit in the Wolfram Language's NSum and NProduct..

Convergence improvement

The improvement of the convergence properties of a series, also called convergence acceleration or accelerated convergence, such that a series reaches its limit to within some accuracy with fewer terms than required before. Convergence improvement can be effected by forming a linear combination with a series whose sum is known. Useful sums include(1)(2)(3)(4)Kummer's transformation takes a convergent series(5)and another convergent series(6)with known such that(7)Then a series with more rapid convergence to the same value is given by(8)(Abramowitz and Stegun 1972).The Euler transform takes a convergent alternatingseries(9)into a series with more rapid convergence to the same value to(10)where(11)(Abramowitz and Stegun 1972; Beeler et al. 1972).A general technique that can be used to acceleration converge of series is to expand them in a Taylor series about infinity and interchange the order of summation. In cases where a symbolic..

Euler transform

There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series(1)into a series with more rapid convergence to the same value to(2)where the forward difference is defined by(3)(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when is taken in the second of Euler's hypergeometric transformations(4)where is a hypergeometric function, it gives Euler's convergence improvement transformation of the series (Abramowitz and Stegun 1972, p. 555).The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21)...

Leibniz criterion

Also known as the alternating series test.Given a serieswith , if is monotonic decreasing as andthen the series converges.

Laplace limit

Let be a complex number, then inequality(1)holds in the lens-shaped region illustrated above. Written explicitly in terms of real variables, this can be written as(2)where(3)The area enclosed is roughly(4)(OEIS A140133).This region can be parameterized in terms of a variable as(5)(6)Written parametrically in terms of the Cartesian coordinates,(7)(8)This region is intimately related to the study of Bessel functions and Kapteynseries (Plummer 1960, p. 47; Watson 1966, p. 270). reaches its maximum value at (OEIS A085984; Goursat 1959, p. 120; Le Lionnais 1983, p. 36), given by the root of(9)or equivalently by the root of(10)as noted by Stieltjes.The minimum value of corresponding to the maximum value is (OEIS A033259; Plummer 1960, p. 47; Watson 1966, p. 270), which is known as the Laplace limit constant. It is precisely the point at which Laplace's formula for solving Kepler's equation begins..

Kummer's test

Given a series of positive terms and a sequence of finite positive constants , let1. If , the series converges. 2. If and the series diverges, the series diverges. 3. If , the series may converge or diverge.The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. With and , the test becomes Raabe's test.

Conditional convergence

A series is said to be conditionally convergent iff it is convergent, the series of its positive terms diverges to positive infinity, and the series of its negative terms diverges to negative infinity.Examples of conditionally convergent series include the alternating harmonic seriesand the logarithmic serieswhere is the Euler-Mascheroni constant.The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge. The Riemann series theorem can be proved by first taking just enough positive terms to exceed the desired limit, then taking just enough negative terms to go below the desired limit, and iterating this procedure. Since the terms of the original series tend to zero, the rearranged series converges to the desired limit. A slight variation works to make the new series diverge to positive infinity or to negative infinity...

Weak convergence

Weak convergence is usually either denoted or . A sequence of vectors in an inner product space is called weakly convergent to a vector in ifEvery strongly convergent sequence is also weakly convergent (but the opposite does not usually hold). This can be seen as follows. Consider the sequence that converges strongly to , i.e., as . Schwarz's inequality now givesThe definition of weak convergence is therefore satisfied.

Comparison test

Let and be a series with positive terms and suppose , , .... 1. If the bigger series converges, then thesmaller series also converges. 2. If the smaller series diverges, then the biggerseries also diverges.

Vitali's convergence theorem

Let be a sequence of functions, each regular in a region , let for every and in , and let tend to a limit as at a set of points having a limit point inside . Then tends uniformly to a limit in any region bounded by a contour interior to , the limit therefore being an analytic function of .

Integral test

Let be a series with positive terms and let be the function that results when is replaced by in the formula for . If is decreasing and continuous for and(1)then(2)and(3)both converge or diverge, where . The test is also called the Cauchy integral test or Maclaurin integral test.

Uniform convergence

A sequence of functions , , 2, 3, ... is said to be uniformly convergent to for a set of values of if, for each , an integer can be found such that(1)for and all .A series converges uniformly on if the sequence of partial sums defined by(2)converges uniformly on .To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms of a uniformly converging series are continuous, then the following conditions are satisfied. 1. The series sum(3)is continuous. 2. The series may be integrated term by term(4)For example, a power series is uniformly convergent on any closed and bounded subset inside its circle of convergence. 3. The situation is more complicated for differentiation since uniform convergence of does not tell anything about convergence of . Suppose that converges for some , that each is differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each..

Herschfeld's convergence theorem

For real, nonnegative terms and real with , the expressionconverges iff is bounded.

Cauchy criterion

A necessary and sufficient condition for a sequence to converge. The Cauchy criterion is satisfied when, for all , there is a fixed number such that for all .

Strong convergence

Strong convergence is the type of convergence usually associated with convergence of a sequence. More formally, a sequence of vectors in a normed space (and, in particular, in an inner product space )is called convergent to a vector in if

Gauss's test

If and given a bounded function of as , express the ratio of successive terms asfor . The series converges for and diverges for (Arfken 1985, p. 287; Courant and John 1999, p. 567).Equivalently, with the same conditions as above, giventhe series converges absolutely iff (Zwillinger 1996, p. 32).

Cauchy condensation test

Let be a series of positive terms with . Then converges iffconverges.

Root test

Let be a series with positive terms, and let1. If , the series converges. 2. If or , the series diverges. 3. If , the series may converge or diverge. This test is also called the Cauchy root test (Zwillinger 1996, p. 32).

Bertrand's test

A convergence test also called "de Morgan's and Bertrand's test." If the ratio of terms of a series can be written in the formthen the series converges if and diverges if , where is the lower limit and is the upper limit.

Riemann series theorem

By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.For example,(1)(2)(3)converges to , but the same series can be rearranged to(4)(5)(6)(7)so the series now converges to half of itself.

Euler's series transformation

Euler's series transformation is a transformation that sometimes accelerates the rate of convergence for an alternating series. Given a convergent alternating series with sum(1)Abramowitz and Stegun (1972, p. 16) define Euler's transformation as(2)where is the forward difference operator(3)and is a binomial coefficient.An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as(4)where is the backward difference operator(5)Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:(6)gives faster convergence,(7)gives same rate of convergence, and(8)gives slower convergence.To see why the Euler transformation works, consider Knopp's convention for difference operator and write(9)(10)Now repeat the process on the series in brackets to obtain(11)and continue to infinity. This proves each finite step in..

Bernoulli's paradox

Suppose the harmonic series converges to :Then rearranging the terms in the sum giveswhich is a contradiction.

Ratio test

Let be a series with positive terms and supposeThen 1. If , the series converges. 2. If or , the series diverges. 3. If , the series may converge or diverge. The test is also called the Cauchy ratio test or d'Alembert ratio test.

Divergent series

A series which is not convergent. Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, rearranging the terms of gives both and .The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series can actually be "summed" rigorously by using extensions to the usual summation rules (e.g., so-called Abel and Cesàro sums). For example, the divergent series has both Abel and Cesàro sums of 1/2...

Radius of convergence

A power series will converge only for certain values of . For instance, converges for . In general, there is always an interval in which a power series converges, and the number is called the radius of convergence (while the interval itself is called the interval of convergence). The quantity is called the radius of convergence because, in the case of a power series with complex coefficients, the values of with form an open disk with radius .A power series always converges absolutely within its radius of convergence. This can be seen by fixing and supposing that there exists a subsequence such that is unbounded. Then the power series does not converge (in fact, the terms are unbounded) because it fails the limit test. Therefore, for with , the power series does not converge, where(1)(2)and denotes the supremum limit.Conversely, suppose that . Then for any radius with , the terms satisfy(3)for large enough (depending on ). It is sufficient to fix a value..

Raabe's test

Given a series of positive terms and a sequence of positive constants , use Kummer's test(1)with , giving(2)(3)Defining(4)then gives Raabe's test: 1. If , the series converges. 2. If , the series diverges. 3. If , the series may converge or diverge.

Absolute convergence

A series is said to converge absolutely if the series converges, where denotes the absolute value. If a series is absolutely convergent, then the sum is independent of the order in which terms are summed. Furthermore, if the series is multiplied by another absolutely convergent series, the product series will also converge absolutely.

Dirichlet's test

Let(1)where is independent of . Then if and(2)it follows that(3)converges.

Abel's uniform convergence test

Let be a sequence of functions. If 1. can be written , 2. is convergent, 3. is a monotonic decreasing sequence (i.e., ) for all , and 4. is bounded in some region (i.e., for all ) then, for all , the series converges uniformly.

Limit test

The limit test, also sometimes known as the th term test, says that if or this limit does not exist as tends to infinity, then the series does not converge. For example, does not converge by the limit test. The limit test is inconclusive when the limit is zero.

Convergent series

A series is said to be convergent if it approaches some limit(D'Angelo and West 2000, p. 259).Formally, the infinite series is convergent if the sequence of partial sums(1)is convergent. Conversely, a series is divergent if the sequence of partial sums is divergent. If and are convergent series, then and are convergent. If , then and both converge or both diverge. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest power terms in polynomials can usually be deleted in both numerator and denominator of a series without affecting convergence.If the series formed by taking the absolute values of its terms converges (in which case it is said to be absolutely convergent), then the original series converges.Conditions for convergence of a series can be determined..

Limit comparison test

Let and be two series with positive terms and supposeIf is finite and , then the two series both converge or diverge.

Abel's convergence theorem

Given a Taylor series(1)where the complex number has been written in the polar form , examine the real and imaginary parts(2)(3)Abel's theorem states that, if and are convergent, then(4)Stated in words, Abel's theorem guarantees that, if a real power series converges for some positive value of the argument, the domain of uniform convergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point.

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