How to find the interval of convergence

Instruction
A power series is a special case of a functional series. It has the form 0 + c1 (z-z0) + c2 (z-z0) ^ 2 + ... + cn (z-z0) ^ n + .... (1) If we make the substitution x = z-z0, then this series takes the form c0 + c1x + c2x ^ 2 + ... + cn (x ^ n) + .... (2)
In this case, rows of the form (2) are more convenient for consideration. Obviously, any power series converges for x = 0. The set of points where the series is convergent (the areaconvergence), can be found based on the Abel theorem. It follows from it that if the series (2) is convergent at the point x0 ≠ 0, then it converges for all x satisfying the inequality | x |
Accordingly, if at some point the x1 row diverges, then this is observed for all x for which | x1 |> | b |. The illustration of Figure 1, where x1 and x0 are chosen large zero, allows us to understand that all x1> x0. Therefore, when they approach, the situation x0 = x1 will inevitably arise. In this case, the situation with convergence, with the passage of merged points (let's call them –R and R), changes in jumps. Since geometrically R is the length, the number R≥0 is called the radius of convergence of the power series (2).Interval(-R, R) is called the convergence interval of a power series. It is also possible R = + ∞. When x = ± R, the series becomes numerical and its analysis is made on the basis of information about the numerical series.
To determine R, the series is investigated for absolute convergence. That is, a series of absolute values ​​of the members of the original series is compiled. Studies can be conducted on the basis of signs of d'Alembert and Cauchy. In their application, limits are found that are compared with the unit. Therefore, a limit of unity is reached at x = R. When deciding on the basis of d'Alembert, the limit presented in fig. 2a


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