1.
Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r

2.
Radius of convergence
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In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a real number or ∞. The radius of r is a nonnegative real number or ∞ such that the series converges if | z − a | < r. The radius of convergence is infinite if the series converges for all complex numbers z, the first case is theoretical, when you know all the coefficients c n then you take certain limits and find the precise radius of convergence. In this second case, extrapolating a plot estimates the radius of convergence, the radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number C = lim sup n → ∞ | c n n | n = lim sup n → ∞ | c n | n | z − a | lim sup denotes the limit superior. The root test states that the series converges if C <1, note that r = 1/0 is interpreted as an infinite radius, meaning that ƒ is an entire function. The limit involved in the ratio test is easier to compute. R = lim n → ∞ | c n c n +1 |, the ratio test says the series converges if lim n → ∞ | c n +1 n +1 | | c n n | <1. That is equivalent to | z − a | <1 lim n → ∞ | c n +1 | | c n | = lim n → ∞ | c n c n +1 |. Usually, in applications, only a finite number of coefficients c n are known. Typically, as n increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity. In this case, two techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio 1 / r where r is the radius of convergence. The basic case is when the coefficients ultimately share a sign or alternate in sign. Domb and Sykes noted that 1 / r = lim n → ∞ c n / c n −1, negative r means the convergence-limiting singularity is on the negative axis. Estimate this limit, by plotting the c n / c n −1 versus 1 / n, the intercept with 1 / n =0 estimates the reciprocal of the radius of convergence,1 / r. This plot is called a Domb–Sykes plot, the more complicated case is when the signs of the coefficients have a more complex pattern. Mercer and Roberts proposed the following procedure, define the associated sequence b n 2 = c n +1 c n −1 − c n 2 c n c n −2 − c n −12 n =3,4,5, …