Infinity and Infinite Series

Transcription

1 Infinity and Infinite Series

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3 Numbers rule the Universe Pythagoras ( BC) God is a geometer Plato ( BC) God created everything by numbers Isaac Newton ( ) The Great Architect of the Universe now begins to appear as a pure mathematician James Jean ( ) Infinity and Infinite Series 3

4 The Sum of Integers ^ OO

5 CHILDREN OFTEN ARGUE with one another over the number of objects that they have. "I have 2 sweets", "I have 3, one more than you". Then they start escalating. "I have 4", "I have 10". "I have 20", "I have 100", and so on. In next to no time, they begin to use bigger numbers, such as thousands, millions, and billions. Soon they run out of all the names of numbers that they know, and they move on to other subjects. The concept of infinity is unknown to children and even to most adults. Indeed it was unknown to many earlier mathematicians who were confused by it, and avoided thinking or talking about the concept. This lasted until about the end of the 16th century. Today mathematicians do not consider infinity (written as ) as a number. The sum of the endless addition of natural numbers does not add up to infinity. Rather, mathematicians say that the sum "tends to infinity", and signify it with the symbol " > ". The dots at the end of the numbers (...) signify the endless continuation of terms. It is easy to understand that if we keep adding up the integers endlessly, the sum gets larger and larger, and hence "tends to infinity". (Mathematicians call natural whole numbers "integers". Once you start calling them integers, you can consider yourself a mathematician too!) Infinity and Infinite Series 5

6 Hie Sum of Factorials 1 + (1 x 2) + (1 x 2 x 3) + (1 x 2 x 3 x 4) + ^ oo 1 + 2! + 3! + 4! +... > oo

7 IF THE SUM of integers tends to infinity, then it is obvious that the sum of the terms, which are the products of the consecutive integers from 1 up to the progressively higher integers indicated, also tends to infinity. This is because while the first two terms are identical to those of the first series of integers, the third and subsequent terms are all greater than the corresponding terms of the first series of integers (e.g., 1 x 2 x 3 is greater than 3). Mathematicians use the term "factorial", represented by (!) as an abbreviation for the product of the consecutive integers from 1 up to the integers indicated (e.g., n\ islx2x3x4x x n).* 'Kathryn, 4, learnt the meaning of n\, and loves to recite it when asked. Infinity and Infinite Series 7

8 The Harmonic Series ^ oo See Proof 14 -> (page 165)

9 IF IT IS obvious that the sum of integers tends to infinity, what about the sum of their reciprocals (opposite page)? First of all, let us note that the bigger the integer, the smaller is the reciprocal. So the individual reciprocals would get smaller and smaller, and tend to zero, as the integers themselves get larger and larger and tend to infinity. Hence, intuitively we would expect the sum of reciprocals to tend to some constant, large though it may be. Surprise, surprise! The sum of the reciprocals of integers also tends to infinity, even though the individual terms tend to zero. Mathematicians refer to a series of terms whose sum tends to infinity as "divergent" "a divergent series". Conversely, a series with a sum that tends to a fixed number (a constant), is referred to as "convergent" "a convergent series". The series (opposite page) enjoys a special place in mathematics and is given the name "the Harmonic Series" because the Greek mathematician Pythagoras and his followers believed that it was related to musical notes. If the harmonic series is divergent, what about the sum of the same series, but with alternating signs: y ^ j + yh? Also, what about the series with the reciprocals of the squares of the integers: J_ J_ J_ J_ J_? l ""' (This last problem had puzzled mathematicians for centuries, until it was solved in the 18th century.) Infinity and Infinite Series 9

10 The Two Halves of the Harmonic Series ^ oo ^ oo See Proof 14» (page 165)

11 IF THE SUM of the reciprocals of all integers tends to infinity, what about the sum of the series which consists of only some of the terms? What about the sum of the reciprocals of all the odd integers? Or the even integers? Again, contrary to intuition, both series (opposite page) are divergent. Yes, the sums of both the series tend to infinity. How many terms of the Harmonic Series do you need to remove before you get a convergent series? This is a problem that continues to puzzle mathematicians. Infinity and Infinite Series 11

12 The Geometric Series = = 2 See Proof 12 -^ (page 156)

13 HAVING SEEN THAT the sums of so many series of reciprocals tend to infinity, it is natural to ask the questions: "Do the sums of all infinite series of reciprocals always tend to infinity?" Are there any infinite series of reciprocals which sum to a constant?" As it turns out, there are numerous infinite series which do sum to constants. "The Geometric Series", an infinite series of some of the reciprocals, sum to a small number, 2. It is amazing how minor differences in the series can sometimes result in dramatic differences in their sums 2 for "the Geometric Series", and "tending to infinity" for "the Harmonic Series" and their odd and even subsets. "The Geometric Series" occupies an important place in the history of mathematics. It was known to the Greeks in the 5th century BC, but it created much confusion, and formed the basis of a famous series of problems known as "Zeno's Paradoxes". Infinity and Infinite Series 13

14 The Exponential Series x2 1x2x3 1x2x3x4, ! + 1! 2! 3! = = e See Proof 22 -> (page 181)

15 THE SUM OF integers and the sum of their reciprocals both tend to infinity. The sum of factorials tend to infinity. So would the sum of the reciprocals of factorials also tend to infinity? Surprise, surprise! It doesn't! It sums to a small number ( ). If we add 1 to the series, we get an extremely important series (opposite page), known as "the Exponential Series". This series was discovered in 1665 by Isaac Newton ( ), the famous English mathematics genius, physicist, astronomer, theologian and polymath. Newton was working at home during a break from his formal studies at Cambridge University, which had to close as a result of a plague from Another famous mathematician, Leonhard Euler ( ) gave the symbol V to the sum of the series (opposite), which sums to V, always written in small letter, turned out to be an important universal constant, featuring in many branches of mathematics, science and technology, from the humble nautilus shell to the immense galaxies in the Universe. Infinity and Infinite Series 15

16 A Logarithmic Series = l 8natural ^ = ln2 = , See Proof 15 > (page 167)

17 ENCOURAGED BY THE two series of reciprocals which sum to constants, one wonders if there are series with alternating terms (i.e. positive terms followed by negative terms, alternately) which sum to constants. For example, the harmonic series, the sum of reciprocals of all the integers, tends to infinity. What about its sister series with alternating terms (opposite)? This series turns out to be a special case of a class of logarithm series, and sums to the natural logarithm of 2 (which is ). Infinity and Infinite Series 17

18 Hie Liebniz-Gregory Series _ 71 = See Proof 40 -» (page 213)

19 ANOTHER SERIES WITH alternating terms is that of the reciprocals of odd integers (opposite page). Remember, the sum of the reciprocals of odd integers tends to infinity. This series of alternating terms is one of the most beautiful series involving it, and is called "the Liebniz-Gregory Series". It sums to a total of -f, and hence serves to introduce us to the wonderful world of the constant it considered by most mathematicians as the most important constant in the world. Infinity and Infinite Series 19

20 The Definition of it n - Circumference of a circle Diameter _C ~D _C^ ~2R

21 THE EARLIEST RECORDS of mankind's awareness of it are to be found among the Babylonians and Egyptians. Some four thousand years ago, they knew about it, as the ratio of the circumference of a circle to its diameter. The Babylonians gave the value of it as -g- > while the Egyptians used (3-), which works out to be irr. It is amazing that the Babylonian value is only 0.5% off the correct value of it, while the Egyptian estimate is 0.6% off. Today, students in elementary schools routinely use the estimate of for it, which despite its simplicity is only 0.04% off its correct value. This estimate is attributed to the great Greek mathematician and physicist, Archimedes ( BC). Yes, he's the one who ran naked in the streets and shouted "Eureka" after he discovered the "principle of displacement" of water. A more accurate approximation of it, but still a simple 335 ratio is H3". This gives it accurate to 6 decimals. it is also essential for the calculation of a number of different properties of curved figures and objects: Area of a circle = itr 2 (R is the radius) Surface area of a sphere = 4itR 2 Volume of a sphere = j 71R Surface Area of a hollow cylinder = 2itRH (H is the height of the cylinder.) Volume of a Cylinder = itr 2 H This simple number, it, has proven to be an extremely important universal constant that finds application in many branches of mathematics, science and technology, beginning with the simple circle and sphere. At the other extreme of the complexity spectrum, it also features in one of Albert Einstein's field equations for his Theory of General Relativity (1916) which described mathematically how the forces of gravity arise out of the curvature of the space-time continuum: I Kb ~ - R gab = GT ab \ Infinity and Infinite Series 21

22 Divergent and Convergent Series > o 2. l!+2!+3! + 4!+5!+6!+--- -> ^«= , ^ o r ^ = =e ( ) 1! 2! 3! 4! 9. i-i + I-i + I-... =ln2 ( ) I-I + I-I + I-... =5( ) THE PLEASURES OF pi,e