When studying functional ranks often use the term of power series which possesses the general member and consists of the whole positive degrees of an independent variable x. During the solution of tasks on this subject it is necessary to be able to find area of convergence of a row.
Instruction
1. Deal with the general concept of convergence. Take some numerical row consisting of the sum of certain parameters and equal to a general meaning. Choose from it a certain interval of n of values which need to be summarized. If with increase of n these sums aspire to a certain final size, then such row is meeting. If sizes increase or decrease infinitely, then in this case a row disperses. For definition of area of convergence of power series use three cases of calculations.
2. Choose any value x from an interval (and; c) power series also substitute it in the general member to reveal absolute convergence. For definition of area of convergence it is necessary to substitute also in the ends of an interval, i.e. x = and and x = century. If the power series are disperse at both values, then the area of convergence is equal (and; в). If the divergence of a row is observed only on the one side of an interval, then the required area is equal [and; c) or (and; c]. For a divergence case on both ends, the piece undertakes [and; c].
3. Check whether the power series meet absolutely at all values x. In this case the interval of convergence and area of convergence will coincide and are equal from "minus" infinity to "plus" infinity.
4. Define that the power series meet only in a point where x =0. According to rules of ranks, in this case the area of convergence will coincide with an interval of convergence and to be equated to zero.
5. Find area of convergence for the set power series. For a start it is necessary to find a convergence interval which pays off, as a rule, on the basis of Dalamber with finding of a limit. It is necessary to make the relation of the following member of power series to previous then to simplify fraction.
6. After that take out also for the sign of a limit together with the sign, and eliminate uncertainty of the relation of infinity. Further the area of convergence of a row is determined by above-mentioned rules.