Higher_Mathematics_Part_3

2.6.Properties of Power Series

1.Power series (2.2) absolutely and uniformly converges on any segment [−a; a] which lies completely within (−R; R).

2.On every interval completely within the interval of convergence the sum of power series is a continuous function.

3.(Differentiation of power series) We can differentiate a power series term-

series γ (x) = a1 + 2a2 x + 3a3 x2 + ... + nan xn−1 + ... obtained by termwise differenti-

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If

∞

S(x) = ∑an xn ,

n=1

then

∞

S′(x) = ∑nan xn−1.

n=1

Theorem If a power series converges in an interval (−R; R), its sum is a

function which has derivatives of any order within the interval of convergence. Each of these derivatives is a sum of the series resulting from term-by-term differentiation of the given series at appropriate number of time. The interval of convergence of each series obtained by differentiation is the same

interval (−R; R).

4. (Integration of power series) If the limits of integration α and β lie within the interval of convergence of a power series then the integral of the sum of the series is equal to the sum of integrals of the terms of series. This series has the

same interval of convergence (−R; R).

has derivatives of all orders. Can we write this function as a power series in powers of x − x0 ? That is,

Let’s find the coefficients of the given series. To do it let’s differentiate a series (2.6) with respect to х:

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

If we substitute x = x0 we get

all orders and remaining member of Taylor’s formula tends to zero if n → ∞ for all x from this interval:

lim Rn (x) = 0 , x (x0 − R; x0 + R).

n→∞

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End of table 2.2

2.9. Some Applications of Power Series

The power series are used for the approximate calculation of values of functions, definite integrals, solutions of differential equations which satisfy initial conditions, and others like.

Thus for alternating series (series of Leibniz’ Type) the following estimation is true:

r(x0 ) = un+1 (x0 ) − un+2 (x0 ) + … < un+1 (x0 ) .

In the case of series with positive terms we can use the following estimation:

r(x0 ) = un+1 (x0 ) + un+2 (x0 ) + … ≤ un+1 (x0 ) + un+2 (x0 ) + … .

∞

If there exists a series ∑ an with positive terms such as

n=1

un+1 (x0 ) ≤ a1, un+2 (x0 ) ≤ a2 , …,

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then

∞

r(x0 ) ≤ ∑ an .

n=1

2.9.2. Approximate Calculation of Definite Integrals

b

Let it is needed to calculate a definite integral ∫ f (x)dx. Suppose an antide-

a

rivative F(x) is not expressed through elementary functions or it is difficult to find it. In this case we make the following actions:

1)set an accuracy of calculation ε > 0 ;

2)expand an integrand f (x) into the power series ( into Maclaurin’s series);

3)use property about term-by-term integration of power series if interval of integration [a; b] belongs to the interval of convergence (−R; R) .

Determination of the error of calculations is the same as for the calculation of values of functions.

2.9.3. Approximate Solution of Differential Equations

Suppose the solution of a differential equation is not expressed through elementary functions or integration of equation is complicated. Then we apply approximate solution of a differential equation using the Taylor’s series.

Suppose it is needed to solve the Cauchy’s problem, that is, to find partial solution of equation

y′′(x0 ). Continuing the process of differentiation and substituting in right part of the received equations the known values, we’ll obtain coefficients y′′′(x0 ),

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y(4) (x0 ) . Taylor’s series with obtained coefficients is the solution of equation

(2.10) for those values x at which it is convergent. The partial sum of this series is the approximate solution of the Cauchy’s problem.

Remark. 1. Considered method is applicable for the solution of differential equations of any order.

2. We do not examine a question concerning the error of approximate solution of differential equation.

Micromodule 2

EXAMPLES OF PROBLEM SOLUTIONS

Find the domain of convergence of functional series.

Therefore, the domain of convergence of this series is x (−∞; − 2) (1; ∞).

∞ n

Example 2. ∑ .

n=1 2nx

Solution. The given series is defined for all real x and the terms of this series are positive. Let’s use d’Alembert’s test:

Therefore, the domain of convergence of the given series is an interval (0; ∞) .

Solution. The given series is defined for all real x except one point x = −1. Let’s consider the following cases:

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We use comparison test of convergence. Then the given series is convergent for x > 1 .

Therefore, the domain of convergence of the given series is (−∞;−1) (1;∞). Investigate the functional series for uniform convergence.

∞ sin nx

Example 4. ∑ .

n=1 n3

Solution. The given series is defined for all real x (−∞; ∞). It is plus-and- minus functional series. Let’s prove this series to be absolutely and uniformly

∞ 1

convergent for all real x. We use Weierstrass’ test. Numerical series ∑

n=1 n3

∞ (−1)n+1

Example 5. ∑ . n=1 n + | x |

Solution. The given series is defined for all real x. It is plus-and-minus functional series. It is an alternating functional series. We’ll estimate a

following inequality is true

rn (x) < ε.

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harmonic series). Consequently, an initial series converges conditionally, but not absolutely.

Find the domain of convergence of power series.

Solution. The given series is a power series. We evaluate the radius of convergence using formula (2.4). We receive

given series is absolutely convergent within the interval (2; 8).

Let’s investigate the behaviour of the series in the end points of interval. Suppose x = 2. Then this series is divergent: