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© Copyright 2002, Jim Loy
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Summation notation: On the left, we see two versions of a series. The left side is the summation notation; the right side is the first few terms of the series. We have a big capital sigma, which means "summation." Then the n=1 means that n starts at one, and increases one at a time. The infinity sign on top means that we never stop increasing n. We would put a number on top, if we wanted n to stop at that number. In all of the series in this article, we write the infinity sign on top. To the right of the sigma is the expression that we are summing for n=1, 2, 3, 4, ... The (-1)^(n+1) (where ^ means take to the power) is a standard device for alternating the sign of the expression; when n=1 then the sign is +; when n=2 the sign is -, etc. Summation notation is more precise, and often clearer than writing out a few terms of the series. Summation notation is just a way of writing several additions, and so it follows the various rules of algebra.
Infinite series: An infinite series is the sum of infinitely many numbers, like 1+2+3+4+... Clearly, this sum is infinite; it is said to diverge. This series trivially sums to zero: 0=0+0+0+0+... It doesn't blow up to infinity, but never gets close to any definite sum either: 1-1+1-1+1-1+... oscillates between 0 and 1, forever, and is said to diverge.
Geometric series: The interesting infinite series have terms which are decreasing and get closer and closer to zero (the terms approach zero as a limit): S=1/2+1/4+1/8+1/16+... This series is a Geometric Series, a series with a common ratio between consecutive terms, and adds up to 1. Another geometric series is S=1/91+1/637+1/4459+1/31213+... The common ratio is 1/7 in this case. The formula for the sum of an infinite geometric series is:
Here a is the first term (1/91) and r is the common ratio (1/7), and the sum S=(1/91)/(6/7)=0.0128205... Another way of writing this series is 1/13(1/7+1/49+1/343+1/2401+...), where every term in the parentheses is a power of 1/7. In other words, the series inside the parentheses is also a geometric series, and we can easily find its sum, with the above formula. We get the same answer. Series which sum to a finite number are said to converge.
Harmonic series: The series 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+... is called the Harmonic Series, and it sums to infinity (very slowly). In other words, it diverges. Similar series, like 1/2+1/4+1/6+1/8+1/10+... are just different versions of the harmonic series, and diverge. In this one, each term is 1/2 times the original harmonic series.
This series also sums to infinity: 1/2+1/3+1/5+1/7+1/11+1/13+... Here the denominators are prime numbers (see the Sieve Of Eratosthenes). It increases very very slowly, but diverges. How about this series: S=1+1/3+1/5+1/7+1/9+1/11... (with odd denominators)? Well, every term is greater than the corresponding term in the 1/prime series, and so it also diverges. In fact the 1/prime series is included in this series (except for 1/2), another clue that it diverges.
Alternating series: And alternating series is one in which every other term is negative. This geometric series alternates: S=1/2-1/4+1/8-1/16+... Every alternating series, in which the terms get closer and closer to zero (as a limit), converge. So alternating series are especially well-behaved. Since this one is a geometric series, we can easily calculate the sum using the formula. An alternating harmonic series would converge: S=1-1/2+1/3-1/4+1/5-... This sum is 0.693147180559945.... This is a famous alternating series: pi/4=1-1/3+1/5-1/7+1/9-..., and is one of the inefficient ways to calculate pi.
With an alternating series, if you have two consecutive partial sums (like Sn and Sn+1), then the actual sum (of infinitely many terms) is between these partial sums. So, not only does a partial sum give an estimate of the actual sum, but it also helps you to determine just how accurate it is.
How about a series like this: S=1/2+1/4-1/8-1/16+1/32+1/64-... (the signs go ++--++--...)? It is not an alternating series, and it is not a geometric series. But it is the sum of two alternating (and geometric) series: 1/2-1/8+1/32-... + 1/4-1/16+1/64, so it converges. Any series which goes ++--++--... can be split up into two alternating series.
Another series: Perhaps this series has a name: S=1/2+1/6+1/12+1/20+1/30+1/42+... The denominators are the product of consecutive integers. In other words, S=1/(1x2)+1/(2x3)+1/(3x4)+1/(4x5)+... (where x means multiplication). The form of the denominators is n(n+1). And 1/(n(n+1))=(1/n)-(1/(n+1)). Perform the subtraction on the right side (by finding a common denominator) to see that that equation is valid. So the sum can be written as S=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+... Getting rid of all of the parentheses, every term subtracts out, except the first term (1). So the sum is 1.
Riemann Zeta Function: The harmonic series (above) is the first Riemann Zeta Function. The second one is where x=2: pi2/6=1+1/22+1/32+1/42+... (the sum of the reciprocals of the squares). Here are the sums of the first few series of this function (the first ten decimal places):
The value approaches 1, as x approaches infinity. As you can see, some of those series can be used to approximate pi.
Some series for approximating pi: These are some of the simplest (and slowest converging) series for estimating pi:
Binomial series: The binomial series is normally not infinite (See The Yanghui Triangle). Well these binomial expansions are infinite:
So, if you come upon something like S=1+2/2+3/4+4/8+5/16+..., then you should recognize it, and be able to evaluate it. Here is a more general infinite binomial expansion:
Notice that this equation uses factorials. n! is called "n factorial." n!=n(n-1)(n-2)...1.
Taylor series and Maclaurin series: I will write about them eventually. Maclaurin series are a simplified form of Taylor series. Many of the series which follow were derived from Maclaurin series.
e: See e. e=1+1/1!+1/2!+1/3!+1/4!+..., which is derived from ex=1+x/1!+x2/2!+x3/3!+x4/4!+...
Trigonometric series:There are many trigonometric series. The three simplest are:
This last one has been used in several different ways to estimate pi.
Hyperbolic series: The two simplest series for hyperbolic functions are:
Fourier series: See other sources for this interesting family of series.
Series reversion (or inversion): If y equals a series involving x, then x can be expressed as a series involving y. See other sources for this interesting subject.
Convergence tests: There are many tests to determine if a series converges. See other sources about that.
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