Web Homework College Algebra

Chapter 5: Elementary Functions

5.1 Infinite Series and Convergence

Throughout this section, we will be dealing with numbers which may either be complex numbers or may be restricted to real numbers. Since the elementary functions will be defined via infinite series, we need to prove some criteria for checking for the convergence of such series.

Definition 1: i. An infinite sequence is a function whose domain is the set of non-negative integers. A infinite sequence is usually denoted as images/elem1.png where images/elem2.png is the value of the function at the non-negative integer i. We say that this infinite sequence converges to a number a (or has limit a) if for every images/elem3.png there is an images/elem4.png such that images/elem5.png for all images/elem6.png . (Intuitively, if images/elem7.png is as close as one wishes to a as soon as i is sufficiently large.) A sequence is said to be convergent if it converges to some number a; otherwise, it is said to be divergent.

An infinite series is an infinite sum images/elem8.png . The images/elem9.png are called the terms of the infinite series. For every non-negative integer n, the images/elem10.png partial sum of the series is images/elem11.png . The infinite series is said to converge to the number a, if the sequence of images/elem12.png partial sums converges to a. A series is said to be convergent or conditionally convergent if it converges to some number; otherwise, it is said to be divergent.

iii. An infinite series images/elem13.png is said to be absolutely convergent if the series images/elem14.png is convergent.

It is important to distinguish between the sequence of terms of an infinite series and the sequence of partial sums of an infinite series.

Proposition 1: If images/elem15.png is convergent, then the sequence images/elem16.png of its terms converges to zero.

Proof: Suppose the series converges to a and that we are given an images/elem17.png . There is an N > 0 such that images/elem18.png satisfies images/elem19.png for all k > N. Suppose that k > N + 1. then k - 1 > N and so images/elem20.png as well as images/elem21.png . Since images/elem22.png , one has by the triangle inequality:

images/elem23.png

as desired.

Example: i. The series images/elem24.png is divergent

ii. If images/elem25.png is an infinite decimal, then the series images/elem26.png converges to the value of the infinite decimal.

iii. A geometric series images/elem27.png converges to images/elem28.png if images/elem29.png . If images/elem30.png , then the geometric series is divergent.

Proposition 2: i. If the series images/elem31.png converges to a and k is a non-negative integer, then the series images/elem32.png converges to images/elem33.png .

ii. Every absolutely convergent series is convergent.

iii. If a series is absolutely convergent, then so is every series obtained from it by re-arranging the order of its terms. Furthermore, reordering the terms does not affect the value to which the series converges.

Proof: i. Let images/elem34.png . Then the partial sums of the original series differ from the partial sums of the second series by b. The result is now a simple consequence of the definitions and the triangle inequality.

ii. Suppose images/elem35.png is absolutely convergent. Let the images/elem36.png be the partial sums of this series. We know that the sum images/elem37.png is convergent; by assertion i of this proposition, it follows that images/elem38.png is also convergent for every non-negative integer k. Further, one has for every n > k,

images/elem39.png

and so we have images/elem40.png in the interval images/elem41.png . Now, it is easy to see that images/elem42.png and that the lengths of these intervals approach zero. So, by Chapter 2, Corollary 8 of section 2.7.2 (or its complex analogue), we know that the intersection of all these intervals contains a single number and the series converges to this value.

iii. For this assertion, it is enough to treat the case where the series consists of non-negative terms. Then the partial sums are form a bounded increasing sequence. Furthermore, every partial sum of any re-arranged series is bounded above by one of the partial sums of this original series. It follows that the partial sums of the re-arranged series also converge.

Proposition 3:(Ratio test) Consider an infinite series images/elem43.png . Suppose there is a real number r with 0 < r < 1 and such that

images/elem44.png

for all sufficiently large k. Then the series is absolutely convergent.

Proof: Because convergence does not depend on the behavior of the first several terms, we can assume that the ratio condition holds for all non-negative k. Further, one can assume all terms are non-negative real numbers. But then we have

images/elem45.png

But then the partial sum images/elem46.png satisfies

images/elem47.png

So, the partial sums form an bounded increasing sequence and therefore converge.

Proposition 4: (Alternating Series Test) Let images/elem48.png for images/elem49.png be a sequence of real numbers which alternate in signs and such that their absolute values satisfy images/elem50.png . If the images/elem51.png approach 0, then the series images/elem52.png is convergent. Furthermore, if the partial sums are images/elem53.png and if the series converges to b, then we have

images/elem54.png

for every non-negative integer k.

Proof: We can assume that the first term of the series is positive. Then it is easy to see that that

images/elem55.png

So, the intervals images/elem56.png are of the type in Corollary 8 of section 2.7.2. So, the partial sums converge to a value b contained in all the intervals. The final inequality follows from the fact that b is in the interval with endpoints images/elem57.png and images/elem58.png .

Example 2: i. The ratio test shows that the series images/elem59.png converges absolutely for all complex numbers z. This is the so-called exponential function exp(z). In particular, the value images/elem60.png is the Euler constant e = 2.718283....

ii. The alternating series test shows that 1 - 1/2 + 1/3 - 1/4 + ... converges. It can be shown that this series is NOT absolutely convergent.

iii. The alternating series exp(-1) = 1 - 1/2! + 1/3! - 1/4! + ... converges very rapidly because of the difference between the limiting value and the images/elem61.png partial sum is no more than 1/(k+1)! in absolute value. In the next section, one will see that exp(-1) = 1/e, and so one gets a very accurate estimate for the Euler constant e.

Proposition 5: Let images/elem62.png and images/elem63.png be convergent series with images/elem64.png partial sums images/elem65.png and images/elem66.png respectively Suppose that the series converge to numbers a and b respectively. Then

  1. The sequence with terms images/elem67.png converges to ab.
  2. If the sequences are absolutely convergent, then so is the series images/elem68.png . Furthermore this series converges to ab.

Proof: i. Let images/elem69.png . We will see below that we need to choose an images/elem70.png such that images/elem71.png and images/elem72.png . Assume that N is chosen so that for all n > N, one has images/elem73.png and images/elem74.png . Then for such n, one has:

images/elem75.png

ii. Apply the first assertion to the series obtained by replacing the terms images/elem76.png and images/elem77.png with their absolute values. One concludes that the sequence with terms images/elem78.png are convergent. Now consider the series images/elem79.png . Since all the terms are non-negative, we really don't care about the order of the terms because if the series converges in one order, it has to converge in all orders. The series will converge if all the partial sums have a common upper bound. But, since the sequence with terms images/elem80.png is convergent, we know that the partial sums do have a common bound. So, our series is absolutely convergent. But then the series images/elem81.png . is also absolutely convergent.

The value to which the series converges does not depend on the order of the terms. So, one might as well order them in an order whose partial sums would be the sequence of images/elem82.png . But then the value of the series is ab by assertion i.

5.2 The Exponential and Logarithm Functions

Definition 2: The exponential function is defined for every complex number z to be the value of the series images/elem83.png .

For example, this means that exp(0) = 1.

Proposition 6: For all complex numbers images/elem84.png and images/elem85.png , one has images/elem86.png

Proof: By Proposition 5, we know that images/elem87.png is absolutely convergent and converges to images/elem88.png . On the other hand, the inside sum is images/elem89.png by the binomial theorem. So, the series is images/elem90.png which completes the proof.

Corollary 1: The exponential function is continuous at every complex number.

Proof: By the addition theorem, one has for every complex number a:

images/elem91.png

In particular, if the exponential function is continuous at 0, then it will be continuous everywhere. Let images/elem92.png . Choose images/elem93.png so that images/elem94.png and images/elem95.png . Then for all z with images/elem96.png , one has

images/elem97.png

So, the exponential function is indeed continuous at 0.

Corollary 2: As a function of real numbers, the exponential function is an increasing function, i.e. exp(x) < exp(y) if x < y.

Proof: By the addition theorem, it is enough to show that 1 = exp(0) < exp(y - x) if y - x > 0. But this is clear from the series since it starts out as 1 + (y - x) and all the remaining terms are positive.

Corollary 3: Again restricting to real values, the range of the exponential function is the set of all positive real numbers. Clearly, the exponential function can be made arbitrarily large by taking x sufficiently large and positive. Since exp(-x) = 1/exp(x) by the addition theorem, it follows that one can make the exponential function arbitrarily small by taking x negative and large in absolute value. Since the exponential function is continuous, the result now follows from the intermediate value theorem.

It follows from Corollary 3 that the exponential function has an inverse function defined for all positive real numbers and with range the set of all real numbers. This function is denoted images/elem98.png is called the logarithm function.

Definition 3: Let a be a positive real number. Then define the power function images/elem99.png for real numbers x.

Corollary 4: For all positive real numbers a and b and real numbers c, one has

  1. images/elem100.png
  2. images/elem101.png
  3. images/elem102.png
  4. images/elem103.png
  5. images/elem104.png is an increasing function.

Proof: These are simple consequences of the definition of the logarithm function as the inverse of the exponential theorem. For example,

images/elem105.png

Applying the logarithm to both sides gives the second assertion.

Similarly, images/elem106.png shows the third assertion.

For the last assertion, suppose that images/elem107.png . Then, since the exponential function is increasing, we have images/elem108.png . But then, using the fact that the exponential function is the inverse of the logarithm function, we get images/elem109.png . Taking the contrapositive, we see that we have shown that if images/elem110.png , then images/elem111.png .

Corollary 5: For all real numbers a, b, r, and s with a and b positive, one has

  1. images/elem112.png
  2. images/elem113.png
  3. images/elem114.png
  4. images/elem115.png

Proof: i. One has

images/elem116.png

Assertion ii is proved similarly.

iii. One has images/elem117.png . So,

images/elem118.png

iv. One has images/elem119.png .

Proposition 7: The natural logarithm function images/elem120.png is continuous at all x > 0.

Proof: The hard part of the proof is to show that images/elem121.png is continuous at x = 1. So, let's assume we have done that part. Let images/elem122.png and let us then show that images/elem123.png is continuous at images/elem124.png . For this, let images/elem125.png and, using the continuity at 1, we know that there is a images/elem126.png such that images/elem127.png provided that images/elem128.png . Let images/elem129.png . Suppose that x is any positive real number such that images/elem130.png . Then images/elem131.png and so images/elem132.png . Since the left side of the inequality is just images/elem133.png , we see that images/elem134.png is continuous at images/elem135.png .

Now, let's try to see why images/elem136.png is continuous at x = 1. Again, let images/elem137.png . Let images/elem138.png be any positive real number smaller than both 1 and images/elem139.png . We will see below that the proper choice of images/elem140.png is a positive number with images/elem141.png . Suppose we choose such a images/elem142.png and then choose any positive x with images/elem143.png . There are two cases to consider:

  1. Suppose x < 1. Then |x - 1| = 1 - x, and so we have

    images/elem144.png

    Solving the inequality, we get images/elem145.png and so images/elem146.png . Since the natural logarithm function is increasing, it follows that images/elem147.png and so

    images/elem148.png

    as desired.

  2. Now, suppose that x > 1. Then |x - 1| = x - 1, and so we have

    images/elem149.png

    because images/elem150.png . Solving, we get images/elem151.png and so applying the logarithm function to both sides, we get

    images/elem152.png

    as desired. This completes the proof of the proposition.

    Proposition 8: The Euler constant e can be represented as a limit:

    images/elem153.png

    Proof: Using the binomial theorem, one has the identity

    images/elem154.png

    So, the basic idea is to show that the additional products in each term do not matter. To see this, choose images/elem155.png and start with

    Lemma 1: If images/elem156.png for images/elem157.png are non-negative real numbers, then images/elem158.png .

    Proof: Clearly, the result is true for m = 1. If it is not true for arbitrary positive integers m, then there would be a smallest positive number m for which it were false. Since m > 1, it would be true for m - 1. So

    images/elem159.png

    Applying Lemma 1, one can estimate the product as:

    images/elem160.png

    where we have used the formula for the sum of an arithmetic series and where the last inequality holds for all n with images/elem161.png . In particular, it holds when images/elem162.png . One has:

    images/elem163.png

    where T is the largest integer less than or equal to images/elem164.png and where we have chosen n large enough so that images/elem165.png .

    5.3 The Trigonometric Functions

    Definition 4: The trigonometric functions are functions of a complex variable z defined by:

    1. images/elem166.png
    2. images/elem167.png
    3. images/elem168.png
    4. images/elem169.png

    The remaining hyperbolic and circular functions are defined in the usual way. These functions can be restricted to real arguments; when we need to distinguish these restricted functions, we will refer to them as the real sine function, real cosine function, etc.

    Letting z = x + iy with x and y real, one has

    images/elem170.png

    Corollary 6: The trigonometric functions are continuous functions such that for all complex numbers images/elem171.png and images/elem172.png :

    1. images/elem173.png
    2. images/elem174.png
    3. images/elem175.png
    4. images/elem176.png and images/elem177.png
    5. images/elem178.png
    6. images/elem179.png
    7. images/elem180.png and images/elem181.png

    Proof: These are trivial to verify using the definitions and the addition formula for the exponential function and its infinite series.

    Proposition 9: i. The real cosine function has a smallest positive root a. We define images/elem182.png .

    ii. images/elem183.png and images/elem184.png .

    Proof: i. For positive real values of x, the series for the real cos(x) function is an alternating series with terms decreasing in absolute value. By the Alternating Series test, one has |cos(x) - 1 + x^2/2| < |x^4|/24. In particular, cos(2) < 0. Since images/elem185.png , Bolzano's Theorem implies that cos(x) has a root between 0 and 2. Since cos(x) is continuous and cos(0) = 1, cos(x) is non-zero near 0.

    Let S be the set of positive roots of the real cosine function. We will do a binary search for the smallest element of S. Starting with the interval images/elem186.png , divide the interval into two equal parts. Discard the left half if it contains no elements of S; otherwise discard the right half. Let a be the real number in the intersection of all the intervals. At each step, the retained interval must contain a point of S and there are no elements of S to the left of the retained interval. In particular, there are roots of the cosine function arbitrarily close to a. By the continuity of the cosine function, it follows that a is itself a positive root of S; so it is the smallest positive root of S.

    With the notation of the Corollary, we have images/elem187.png and so images/elem188.png . One has images/elem189.png and so images/elem190.png . This tells us that images/elem191.png . Repeating, we get images/elem192.png and images/elem193.png .

    From the addition formulas, one now has

    images/elem194.png

    and

    images/elem195.png

    which completes the proof.

    Remark: The alternating series for the sine function shows that images/elem196.png is positive between 0 and 2. But then images/elem197.png . We know now that both the cosine and sine function are positive between 0 and images/elem198.png . One can now calculate the values of the trig functions at some simple angles:

    1. When x = images/elem199.png , the double angle formula for cosine shows that images/elem200.png . But we also have images/elem201.png . Since the both functions are positive there, we have images/elem202.png .
    2. Now try x = images/elem203.png . One has images/elem204.png This shows that images/elem205.png and so images/elem206.png . But then images/elem207.png and images/elem208.png .
    3. Finally, try x = images/elem209.png . Using images/elem210.png and so images/elem211.png and so images/elem212.png .

    Remark: One can use the double angle formulas to show that the cosine and sine functions at images/elem213.png agree with those defined using the unit circle. But then multiple angle formulas show that the two definitions coincide on a dense set of rational multiples of images/elem214.png between 0 and images/elem215.png . By continuity, it follows that the definitions of the trig functions using series corresponds with the definition using the unit circle.

    Another more direct approach can be taken. We need:

    Proposition 10: The real cosine function is decreasing on the interval from 0 to images/elem216.png .

    Proof: This follows from the addition formula for cosines assuming that we are only interested in the interval from 0 to images/elem217.png . To double the interval, use images/elem218.png which also is a consequence of the addition formula.

    Since the cosine function is decreasing on the interval from 0 to images/elem219.png , the function is one-to-one. So, it has an inverse function, which we call the arccosine or inverse cosine function. Since the cosine function is continuous and decreases from 1 to -1 between 0 and images/elem220.png , it follows that the the graph passes through every inermediate point. Therefore, the inverse function is defined for every real number between 1 and -1. For any non-zero complex number z, let images/elem221.png . Then u has length 1 and so a lies between -1 and 1. So, its arccosine is defined. We define images/elem222.png where sgn = 1 if b is non-negative and -1 otherwise. This has been defined so that one has

    images/elem223.png

    So, one concludes that the sine and cosine functions defined in this chapter correspond with the ones defined via the unit circle.

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    Revised: June 22, 2001
    All contents © copyright 2001 K. K. Kubota. All rights reserved