Solution of IDZ Ryabushko 12.2 - Option 19

Section: Chapter 12. Series
Topic: Domain of convergence of power series
Description: Finding the domain of convergence of power series: determining the radius of convergence using d'Alembert's and Cauchy's formulas, investigating convergence at the endpoints of the interval. Also includes expansion of functions into Maclaurin and Taylor series.

Tasks:
1. Find the domain of convergence of the series (1–3)
2. Expand the function f(x) into a Maclaurin series. Indicate the domain of convergence of the obtained series to this function (4)
3. Calculate the indicated quantity approximately with the given degree of accuracy α, using the expansion into a power series of an appropriately chosen function (5)
4. Using the expansion of the integrand into a power series, calculate the indicated definite integral with an accuracy of 0.001 (6)
5. Find the expansion into a power series in powers of x of the solution of the differential equation (write down the first three terms of this expansion different from zero) (7)
6. Using the method of successive differentiation, find the first k terms of the expansion into a power series of the solution of the differential equation under the given initial conditions (8)

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