Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 12



Power series

Definition  

Let be a sequence of real numbers and another real number. Then the series

is called the power series in for the coefficients .

For a power series, it is important that varies and that the power series represents in some convergence interval a function in . Every polynomial is a power series, but one for which all coefficients starting with a certain member are . In this case, the convergence is everywhere.

We have encountered an important power series earlier, the geometric series (here all coefficients equal ), which converges for and represents the function , see Theorem 9.13 . Another important power series is the exponential series, which for every real number converges and represents the real exponential function. Its inverse function is the natural logarithm.

The behavior of convergence of a power series is given by the following theorem.


Theorem

Let

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be a power series and suppose that there exists some such that converges. Then there exists a positive (where is allowed) such that for all fulfilling the series converges absolutely. On such an (open) interval of convergence, the power series represents a continuous function.

Proof

The proof needs a systematic study of power series and of limits of sequences of functions. We will not do this here.


If two functions are given by power series, then their sum is simply given by the (componentwise defined) sum of the power series. It is not clear at all by which power series the product of two power series is described. The answer is given by the Cauchy-product of series.


Definition  

For two series and of real numbers, the series

is called the Cauchy-product of the series.

Also, for the following statement we do not provide a proof.


Lemma

Let

be absolutely convergent series of real numbers. Then also the Cauchy product is absolutely convergent and for its sum the equation

holds.


From this we can infer that the product of power series is given by the power series whose coefficients are those which arise by the multiplication of polynomials, see Exercise 12.3 .



Exponential series and exponential function

We discuss another important power series, the exponential series and the exponential function defined by it.


Definition  

For every , the series

is called the exponential series in .

So this is just the series


Theorem

For every , the exponential series

is

absolutely convergent.

Proof  

For , the statement is clear. Else, we consider the fraction

This is, for , smaller than . By the ratio test we get convergence.


Due to this property, we can define the real exponential function.

The graph of the real exponential function

Definition  

The function

is called the (real)

exponential function.

The following statement is called the functional equation for the exponential function.


Theorem

For real numbers , the equation

holds.

Proof  

The Cauchy product of the two exponential series is

where

This series is due to Lemma 12.4 absolutely convergent and the limit is the product of the two limits. Furthermore, the -th summand of the exponential series of equals

so that both sides coincide.



Corollary

The exponential function

fulfills the following properties.
  1. .
  2. For every , we have . In particular .
  3. For integers , the relation holds.
  4. For every , we have .
  5. For we have , and for we have .
  6. The real exponential function is strictly increasing.

Proof  

(1) follows directly from the definition.
(2) follows from

using Theorem 12.8 .
(3) follows for from Theorem 12.8 by induction, and from that it follows with the help of (2) also for negative .
(4). Nonnegativity follows from


(5). For real we have , so that because of (4), one factor must be and the other factor must be . For , we have

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as only positive numbers are added.
(6). For real , we have , and therefore, because of (5) , hence


With the help of the exponential series, we also define Euler's number.


Definition  

The real number

is called Euler's number.

So we have . Its numerical value is


Remark

For Euler's number there is also the description

so that can also be introduced as the limit of this sequence. However, the convergence in the exponential series is much faster.

We will write also instead of . This is, for , compatible with the usual meaning of powers in the sense of the fourth lecture due to Corollary 12.9 . The compatibility with arbitrary roots (if the exponents are rational) follows from Remark 12.17 and Exercise 12.17 .


Theorem

The real exponential function

is

continuous and defines a bijection between and .

Proof  

The continuity follows from Theorem 12.2 , since the exponential function is defined with the help of a power series. Due to Corollary 12.9 , the image lies in , and the image is, because of the intermediate value theorem an interval. The unboundedness of the image follows from Corollary 12.9 . This implies, because of Corollary 12.9 , that also arbitrary small positive real numbers are obtained. Thus the image is . Injectivity follows from Corollary 12.9 , in connection with Exercise 5.38 .




Logarithms

Definition  

The natural logarithm

is defined as the inverse function of the

real exponential function.

Theorem

The natural logarithm

is a

continuous strictly increasing function, which defines a bijection between and . Moreover, the functional equation

holds for all .

Proof  

The exponential functions for various bases

Definition  

For a positive real number , the exponential function for the base is defined as


Theorem

Let denote a positive real number. Then the exponential function

fulfills the following properties.
  1. We have for all .
  2. We have .
  3. For and , we have .
  4. For and , we have .
  5. For , the function is strictly increasing.
  6. For , the function is strictly decreasing.
  7. We have for all .
  8. For , we have .

Proof



Remark

There is another way to introduce the exponential function to base . For a natural number , one takes the th product of with itself as definition for . For a negative integer , one sets . For a positive rational number , one sets

where one has to show that this is independent of the chosen representation as a fraction. For a negative rational number, one takes again the inverse. For an arbitrary real number , one takes a sequence of rational numbers converging to , and defines

For this, one has to show that these limits exist and that they are independent of the chosen rational sequence. For the passage from to , the concept of uniform continuity is crucial.


Definition  

For a positive real number , , the logarithm to base of is defined by

Logarithms for various bases

Theorem

The logarithms to base

fulfill the following rules.
  1. We have and , this means that the logarithm to Base is the inverse function for the exponential function to base .
  2. We have .
  3. We have for .
  4. We have

Proof


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