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Series Formulas
Sum of first n positive integers
The series:
S = 1 + 2 + 3 + 4 + ... + n
Is a fairly useful one. Its closed form value is given by:
This can be used to derive a number of related formulas. For example, the series:
S = m + m+1 + m+3 + m+4 + ... + n
can be thought of as a combination of two series: the first consisting of the integers from 1 to n and the second of the integers from 1 to m−1.
That is:
S = [1 + 2 + 3 + ... + m−1 + m + m+1 +... + n] − [1 + 2 + ... + m−1]
To get the desired sum we subtract the "1 to m−1" sum from the "1 to n" series:
| | S = | n(n+1) | − | (m−1)(m−1+1) | = | n(n+1) − m(m-1) |
2 | 2 | 2 |
Another possibility is the sum of the first n even integers:
Seven = 2 + 4 + 6 + 8 + ... + 2n
We can get an expression for the sum by factoring a 2 out of each term:
Seven = 2[1 + 2 + 3 + 4 + ... + n]
The piece in the brackets is just the sum of the first n integers with a sum given by our formula above. Therefore, we get:
Seven = 2n(n+1)/2 = n(n+1)
The sum of the first odd integers also can be teased out:
Sodd = 1 + 3 + 5 + ... + (2n−1)
This series has exactly n terms. If we add 1 to each term, we will have added a total of n so we can keep everything equal if we also subtract n.
Sodd = (1+1) + (1+3) + (1+5) + ... + [1+(2n−1)] − n
= [2 + 4 + 6 + ... + 2n] − n
The series with the brackets is just the sum of the first n even integers. Therefore the total becomes the rather remarkable formula:
Sodd = n(n+1) − n = n2 + n − n = n2
Sum of squares of first n positive integers
The series:
Ssquares = 12 + 22 + 32 + 42 + ... + n2
= 1 + 4 + 9 + 25 + ... + n2
also can be evaluated, but since it is not a simple arithmetic series, the process is more difficult. The result is given by:
| | Ssquares = | n(n+1)(2n+1) |
6 |
Sum of cubes of first n positive integers
The series:
Scubes = 13 + 23 + 33 + 42 + ... + n3
= 1 + 8 + 27 + 64 + ... + n3
also can be evaluated, but since it is not a simple arithmetic series, the process, like the one for the sum of the squares, is more difficult. The result is given by
a particularly interesting formula:
Note that this is the square of the formula for the sum of the first n integers!
Sum of higher powers of first n positive integers
We can continue with higher degree powers. Unfortunately, there is no
simple pattern to the formulas and each higher power involves more work to obtain a closed form.
The series:
S4 = 14 + 24 + 34 + 44 + ... + n4
= 1 + 16 + 81 + 256 + ... + n4
becomes
| | S4 = | n(n+1)(2n + 1)(3n2 + 3n − 1) |
30 |
The series:
S5 = 15 + 25 + 35 + 45 + ... + n5
= 1 + 32 + 243 + 1024 + ... + n5
evaluates to
| | S5 = | n2(n+1)2(2n2 + 2n − 1) |
12 |
The series:
S6 = 16 + 26 + 36 + 46 + ... + n6
= 1 + 64 + 729 + 4096 + ... + n6
can be computed from
| | S6 = | n(n+1)(2n + 1)(3n4 + 6n3 − 3n + 1) |
42 |
The series:
S7 = 17 + 27 + 37 + 47 + ... + n7
= 1 + 128 + 2187 + 16384 + ... + n7
is
| | S7 = | n2(n+1)2(3n4 + 6n3 − n2 − 4n + 2) |
24 |
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© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2008 by Lawrence Turner |