2.6. Properties of power series

Property 1. A power series uniformly converges on each interval , where and is the radius of convergence.

The proof. Consider the interval . We will have

For all that meet the condition and for . But since the number series converges, by the Weierstrass’s test, the fiven series converges uniformly on the interval .

Property 2. The sum of series is continues inside its interval of convergence.

The proof. Any point in the convergence interval can be included in the interval where the series converges uniformly and its terms are continues.

Property 3. Term-by-term differentiation or integration of power series does not change its radius of convergence.

The proof. Let R be the radius of convergence of the series

Denote by the sum of the series for which

Thus we have the same interval of convergence. Making use theorem 3 we obtain

Property 4. The relation can be termwise integrated and differentiated any number of times.

The proof. By applying Theorem 3., to the series containing the first-order derivatives and then to the series containing the second-order derivatives, we get the formula

for any .

In particular, properties 4 and 2 imply that the sum of a power series possesses continuous derivatives of all the orders within its interval of convergence.

Example 2.10. Let us take a series:

(we apply formula for the sum of an infinite geometric progression:

.

Integrating termwise we obtain:

The series on the right-hand side is convergent for as well. Hence we have managed to find the sum of an interesting series of the form:

.

Performing termwise integration and differentiation of a given series we can sometimes reduce the series to a series whose sum is known and thus find the sum of the series in question.

2.7. Taylor’s series of elementary functions

Let a function be defined and have derivatives of all orders in some neighborhood of a point .

Then the power series

Is called the Taylor’s series of the functions at the point . Note, that when , the Taylor’s series takes a simpler form, namely

And this is called the Maclaurin series of about .

Lemma. In order for a Taylor’s series of the function to converges to at a point , it is necessary and sufficient that at this point the remainder of Taylor’s formula tends to zero as .

The proof. From the Teylor’s formula:

It follows that if as , then

That is, Taylor’s series of the function converges to .

Theorem. The Taylor’s series of the functions and converge to them for any

.

Let us prove the first formula. The radius of convergence of the series

So that it converges everywhere on the number line. The Taylor’s formula with a remainder in Lagrange’s form for the function has the form:

,

Here c lies between and . According to the lemma we have to prove that for any fixed .

To find the limit we consider the series And use the ratio test:

So that the necessary condition of convergence gives us for any fixed .

The other two formulas are proved in a similar way.