Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
Infinite Series and Partial Sums
The summer fascination is the World Series. Not an infinite series, although it does seem to last forever, especially when we're heading into the eighth or ninth inning.
The idea of a series appears in math, but unlike a succession of baseball games, we have a sequence of terms. These terms are numbers, and the sequence means the numbers follow a pattern. The sum of terms in an infinitely long sequence is an infinite series. Summing part of the sequence is called a partial sum. Let's explore infinite series and partial sums a little more by looking at the different types.
If we have a sequence with the first three terms of 1, 3, and 5, what is the next number? The answer is 7. Why? The difference between successive terms is 2. Thus, the next number is 5 + 2 = 7. When the difference between successive terms in a series is constant, the sequence is an arithmetic series.
The arithmetic series 1 + 2 + 3 + … is shown here.
The capital sigma means to sum numbers. The expressions below and above the sigma tell us the 'from' and the 'to' for the sum. This sum runs from n = 1 to n = infinity. The n = above the sigma is understood and not repeated. The sum of terms keeps getting longer, approaching infinity. But what about a partial sum? For example:
1 + 2 + 3 + 4 + 5 + 6 = 21
(1 + 6) + (2 + 5) + (3 + 4)
Each parenthesized sum equals 7, and there are 3 sums, and 7 times 3 is 21! Adding the first term (the 1) plus the last term (the 6) gives 7, and multiplying the last term (the 6) divided by 2 gives 3.
As you can see, with this notation, this result works for both n even or odd and really pays off for large numbers. Imagine having to add 1 to 1000 one number at a time?
Put simply, a geometric series has a constant ratio of successive terms.
Extend this sequence:
1/2, 1/4, and 1/8
1/2 divided by 1/4 is 2. 1/4 divided by 1/8 is 2. The next term in the sequence is 1/8 divided by 2 = 1/16. In this example, each term is 1/2 raised to increasingly high powers. 1/4 is (1/2)^2; 1/8 is (1/2)^3; and 1/16 is (1/2)^4.
As you can see in the above equation, the equation is using the letter a for the 1/2. This sum starts at n = 0 with a^0 = 1. The sum is 1 / (1 - a) if the absolute value of a < 1. Otherwise, the sum is infinite.
How about a geometric series with a = 1/2? The sum is 1 / (1 - 1/2) = 2.
How about the geometric series 1 - 1/2 + 1/4 - 1/8 + … ?
As you can see, in this series, a = -1/2. The absolute value of -1/2 is 1/2. The sum is 1 / (1 - (-1/2)) = 1 / (1 + 1/2) = 1 / (3/2) = 2/3.
A series with 1 over a harmonic, which is an integer multiple of a fundamental number, is the harmonic series. If the fundamental number is 1, the harmonics are 2, 3, 4, ….
Although each successive term is getting smaller, the sum carried out to infinity is infinite.
Alternating the signs of the harmonic series gives the alternating harmonic series:
Unlike the harmonic series, the alternating harmonic series has a derivable finite sum, which you can see reflected in the calculations.
Given x, -(1/2)x^2 and (1/3)x^3, what is the next term?
Take a look at the series below:
ln is the natural logarithm, and this series is valid for -1 < x < or = 2.
Letting x = 1 and simplifying:
ln(1 + 1) = ln(2), showing the alternating harmonic series equals ln(2).
A telescopic series has finite terms after cancelling terms in its partial sum. Here is an example:
This sum ends at n = N. Although this is a partial sum, it doesn't yet show the 'telescopic' nature. This is where the clever trick below comes in:
Thus, 1/2 is 1 - 1/2, 1/6 is 1/2 - 1/3 and 1/12 is 1/3 - 1/4.
Thus, 1/(n(n+1)) is 1/n - 1/(n+1).
Expand the partial sum:
The cancelled terms 'telescope' down the sum.
This partial sum ends at n = N and the resulting sum is 1 - 1/(N+1). If N goes to infinity, the partial sum becomes an infinite series. As N gets larger and larger, 1/(N+1) gets smaller and smaller. In the limit as N approaches infinity, 1/(N+1) is zero. This particular telescopic series equals 1.
Let's briefly recap the main terms that we used to learn what we've learned here.
Organized lists of number terms constitute a sequence. An infinite series sums all the terms in an infinitely long sequence. Summing only a portion is a partial sum.
An arithmetic series has a constant difference between successive numbers, while it's the ratio of successive numbers that is constant for a geometric series. A harmonic series has fractional numbers whose denominator increases harmonically. The alternating harmonic series is a harmonic series with alternating signed terms. When the partial sum of a telescopic series is calculated, the length of the sequence shortens.
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