Mathematics
Radius of Convergence
The radius of convergence is a concept in power series that determines the interval of values for which the series converges. It is the distance from the center of the series to the nearest point where the series diverges. The radius of convergence is determined by the coefficients of the series and can be calculated using various methods.
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7 Key excerpts on "Radius of Convergence"
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
The question of the behaviour of the series on the Radius of Convergence is usually a very difficult one to answer, and varies wildly from series to series; anything can happen (Fig. 3.7). Convergence everywhere inside circle Convergence must be checked for each individual point D ivergence outside circle Fig. 3.7 The circle of convergence for a complex series. 148 A Mathematical Bridge 3.2.3.1 Determining the Radius of Convergence We can use the ratio test to try to determine the value of the Radius of Convergence for a power series S = ∑ ∞ n =0 a n z n : • lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 z n +1 a n z n vextendsingle vextendsingle vextendsingle = lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle | z | braceleftBigg < 1 ⇒ convergence = 1 ⇒ ? > 1 ⇒ divergence Although it is an obvious point, it is noteworthy that the fixed value of the limit of | a n +1 /a n | is independent of the choice of z . Suppose that this has some well defined limit. We can then use this to determine the Radius of Convergence • 1 R = lim n →∞ vextendsingle vextendsingle vextendsingle a n +1 a n vextendsingle vextendsingle vextendsingle (if the limit exists and is positive) As a special case, we define R = ∞ if the limit of the ratio is zero. This means that the power series converges for any value of z . At the other end of the spectrum, if the ratio of the limit is infinite then the Radius of Convergence of the series is zero: the series will diverge for any non-zero choice of z . 3.2.4 Rearrangement of infinite series We have really begun to make progress into the theory of limiting processes, and now have clear guidelines concerning the questions of convergence or divergence of a given infinite sequence.
- Bernhard W. Bach, Jr.(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
The constant r is (if you haven ’ t already guessed) referred to as the Radius of Convergence. Equations of the form j z z 0 j < r ; occur frequently, so it is worth memorizing that such an equation describes a disk in the complex plane. To recognize that the equation j z z 0 j < r describes the interior of a circle , rewrite the complex variable z and the complex constant z 0 in rectangular form, so that z ¼ x þ iy and z 0 ¼ a þ ib : Substituting these expressions for z and z 0 in the equation for the disk, we fi nd that j z z 0 j ¼ jð x þ iy Þ ð a þ ib Þj ¼ jð x a Þ þ i ð y b Þj < r : Recall that the absolute value or length of a complex number is writing as j z j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 p , so that jð x a Þ þ i ð y b Þj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x a Þ 2 þ ð y b Þ 2 q < r : Squaring both sides of this expression, we get ð x a Þ 2 þ ð y b Þ 2 < r 2 : This inequality describes the interior of a circle of radius r centered at the point ( x, y ) = ( a, b ); see Figure 4.15 . In other words, a complex series converges for values of z near the center ( a, b ) of a disk and diverges when z is too far away. The Radius of Convergence r speci fi es how far one can stray from the center of the disk and still expect the 124 Complex In fi nite Series series to converge. If a series converges for all values of z , then the Radius of Convergence is in fi nite. Note that we have not addressed the convergence proper-ties of a series on the boundary j z z 0 j ¼ r : The reason for this is that the behavior of a series on a boundary can be very complicated. Recall that for ρ = 1, the ratio test is inconclusive; the series may be convergent or divergent. This means that it is possible for a series to be convergent at some locations on the boundary and divergent at others.
eBook - PDF- Hemant Kumar Pathak, Ravi Agarwal, Yeol Je Cho(Authors)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
By Theorem II, we see that the Radius of Convergence R of a power series 128 Functions of a Complex Variable is given by 1 R = lim sup n →∞ | a n | 1 n . This is known as Hadamard’s Formula for the Radius of Convergence. In practice, we use a simpler formula for finding R . It is given by 1 R = lim n →∞ a n +1 a n . (3 . 3) The formula (3.3) follows from (3.2) and Cauchy’s Second Theorem on limits which states that if { u n } is a sequence of positive constants, then lim n →∞ ( u n ) 1 /n = lim n →∞ u n +1 u n provided the limit on the R.H.S. exists whether finite or infinite. Note that Theorem III does not make any statement on the boundary of the circle | z | = R . Thus, a power series may converge or diverge on the boundary. In view of the above definition, Theorem III may be stated as: A power series converges absolutely and uniformly within its circle of con-vergence and diverges outside it. Theorem IV. The power series ∞ ∑ n =0 na n z n -1 , obtained by differentiating the power series ∞ ∑ n =0 a n z n , has the same Radius of Convergence as the original series. Proof. Let R and R 0 be the radii of convergence of the series ∞ ∑ n =0 a n z n and ∞ ∑ n =0 a n z n -1 , respectively. Then we have 1 R = lim sup n →∞ | a n | 1 /n , 1 R 0 = lim n →∞ n 1 /n | a n | 1 /n . In order to prove R = R 0 , we have to establish that lim n →∞ n 1 /n = 1 . By Cauchy’s second theorem on limits, we have lim n →∞ a 1 /n n = lim n →∞ a n +1 a n . Therefore, we have lim n →∞ n 1 /n = lim n →∞ n + 1 n = lim n →∞ 1 + 1 n = 1 . Hence R = R ’. This completes the proof. Power Series and Elementary Functions 129 3.2.4 Analyticity of Sum Functions of Power Series If f ( z ) = ∞ ∑ n =0 a n z n , then f ( z ) is called the sum function of the power series ∞ ∑ n =0 a n z n . Now, we prove the following important theorem on the analytic character of the sum function. Theorem V. The function f ( z ) of the series ∞ ∑ n =0 a n z n represents an analytic function inside its circle of convergence.
eBook - PDF- E. T. Whittaker, G. N. Watson, Victor H. Moll(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
Let lim | a n | -1/n = r ; then, from §2.35, ∞ ˝ n=0 a n z n converges absolutely when | z | < r ; if | z | > r , a n z n does not tend to zero and so ∞ ˝ n=0 a n z n diverges (§2.3). The circle | z | = r , which includes all the values of z for which the power series a 0 + a 1 z + a 2 z 2 + a 3 z 3 + · · · converges, is called the circle of convergence of the series. The radius of the circle is called the Radius of Convergence. In practice there is usually a simpler way of finding r , derived from d’Alembert’s test (§2.36); r is lim(a n /a n+1 ) if this limit exists. A power series may converge for all values of the variable, as happens, for instance, in the case of the series 5 z - z 3 3! + z 5 5! - · · · , which represents the function sin z; in this case the series converges over the whole z-plane. On the other hand, the Radius of Convergence of a power series may be zero; thus in the case of the series 1 + 1! z + 2! z 2 + 3! z 3 + 4! z 4 + · · · we have | u n+1 /u n | = n| z | , which, for all values of n after some fixed value, is greater than 5 The series for e z , sin z , cos z and the fundamental properties of these functions and of log z will be assumed throughout. A brief account of the theory of the functions is given in the Appendix. 2.6 Power series 29 unity when z has any value different from zero. The series converges therefore only at the point z = 0, and the radius of its circle of convergence vanishes. A power series may or may not converge for points which are actually on the periphery of the circle; thus the series 1 + z 1 s + z 2 2 s + z 3 3 s + z 4 4 s + · · · , whose Radius of Convergence is unity, converges or diverges at the point z = 1 according as s is greater or not greater than unity, as was seen in §2.33. Corollary 2.6.1 If (a n ) be a sequence of positive terms such that lim(a n+1 /a n ) exists, this limit is equal to lim a 1/n n .
eBook - PDFIntroduction to Real Analysis
An Educational Approach
- William C. Bauldry(Author)
- 2011(Publication Date)
- Wiley(Publisher)
Taylor's theorem on the expansion of a function in a power series was first discovered by Gregory in 1671, although in a different form using differences. [See Stillwell (1989, Chapter 9).] POWER SERIES AND TAYLOR SERIES 31 Power Series Generalizing the geometric series £) ar n by replacing r with a variable and a with a sequence {a n } is a natural step. Definition 1.17 (Power Series) A power series centered at c is a series having the form oo ^2a n (x - c) n = a 0 +ai(x - c) + a 2 (x - c) 2 H 71=0 where the a n are constants. By convention, ÜQ(X — c)° = oo for all values of x even though 0° is indeterminate. Suppose a power series converges for some x that is R units from c; then the comparison test tells us that the series converges for any x closer to c, i.e., for all x with x — c < R. Prove it! Similarly, if a power series diverges for some x that is R units from c, then it diverges for all x with x — c > R. This observation leads us to define the Radius of Convergence of a power series. Definition 1.18 (Radius of Convergence) For a power series^ a n{x — c) n , exactly one of the three following statements must be true. 1. The series converges only at x = c. The Radius of Convergence is R = 0. 2. The series converges for all x — c < R and diverges for all jrr — c| > Rfor some positive value R. The Radius of Convergence is R. 3. The series converges for all x. The Radius of Convergence is R = oo. Given a positive Radius of Convergence R for the power series ^a n (x — c) n , the interval of convergence is the interval from c — R to c + R that may or may not contain the endpoints—they must both be checked. Often, the Radius of Convergence can be found using the ratio or root tests. However, both tests fail at an endpoint of the interval of convergence, and other methods are required. Mercator's expansion of ln(l + x) comes from integrating the geometric series expansion for 1/(1 + x).
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon, M. Stark, S. Ulam(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
In the latter case, whatever value of χ we take, it is necessary to find an χ such that | x | < | ^ | , and then, by the lemma, for the value of χ taken the series converges absolutely. The series is every-where convergent. Now let the set {|3c|} be bounded above, and let R be its upper bound (so that 0 < < oo). If x > then this value of χ is known to differ from all x, and the series diverges. Now let us take any X for which | x | < i ^ . By the definition of a bound [Sec. 6], it is necessary to find an χ such that < |3c| but by the lemma this again implies the absolute convergence of the series (1). Thus we have proved the general THEOREM. For every power series (1), provided only that it is not everywhere divergent, there exists a positive number R (it may also be + Qo) such that the series converges absolutely for |Λ:| < JR, and the series diverges forx>R (if R< oo). This number R is called the Radius of Convergence of the series. 8 8 16. SEQUENCES AND SERIES OF FUNCTIONS n = l R = CO, the interval of convergence is ( - oo, + oo) [Sec. 253]. (2) In the case of the progression 00 1 + y] X« n = l = 1, the interval of convergence is ( - 1, + 1): both end points are excluded. (3) The series has = 1, the interval of convergence is [ -1 , +1]: both end points are included, but there the convergence is non-absolute [Sec. 255]. (4) For the series x^ Σ<-»-ί R = 1, the interval of convergence is ( — 1, +1]: the left end point is not included, but at the right end point the series converges non-absolutely [Sec. 256]. (5) Finally, we consider the series ^ χη n = l here also R = , and the interval of convergence is [— 1, +1], the series converges ( 00^ 1 in view of the convergence of the series — ) .
eBook - PDF- A.K. Katsaras, W.H. Schikhof, L. Van Hamme(Authors)
- 2001(Publication Date)
- CRC Press(Publisher)
Convergence on the Levi-Civita Field and Study of Power Series * Khodr SHAMSEDDINE Department of Mathematics and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, email: [email protected] Martin BERZ Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, email: [email protected] Abstract Convergence under various topologies and analytical properties of power series on Levi-Civita fields are studied. A Radius of Convergence is established that as-serts convergence under a weak topology and reduces to the conventional Radius of Convergence for real power series. It also asserts strong (order) convergence for points the distance of which from the center is infinitely smaller than the Radius of Convergence. In addition to allowing the introduction of common transcendental functions, power series are shown to behave similar to real power series. Besides being infinitely often differentiable and re-expandable around other points, it is shown that power series satisfy a general intermediate value theorem as well as a maximum theorem and a mean value theorem. 1991 Mathematics Subject Classification. 26E30, 12J25 Key words and phrases. Levi-Civita field, power series, weak convergence, strong conver-gence, calculus 1 Introduction It is a known fact that topological continuity or differentiability of a function on a closed interval of any non-Archimedean field are not sufficient to guarantee that the function assumes all the intermediate values nor a maximum nor a minimum on the interval. These problems are common to all non-Archimedean sructures and are due to the total disconnectedness of these structures in the order topology. It is shown in [1] that under stronger definitions for continuity and differentiability and under mild conditions, the *This research was supported by an Alfred P. Sloan fellowsl!ip and by the United States Department of Energy, Grant # DE-FG02-95ER40931. 283
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