Relationships to other operations and constants
It's possible to add fewer than 2 numbers.
If you add the single term x, then the sum is x.
If you add zero terms, then the sum is zero, because zero is the identity for addition.
This is known as the empty sum.
These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case.
For example, if m = n in the definition above, then there is only one term in the sum; if m = n + 1, then there is none.
Many other operations can be thought of as generalised sums.
If a single term x appears in a sum n times, then the sum is nx, the result of a multiplication.
If n is not a natural number, then the multiplication may still make sense, so that we have a sort of notion of adding a term, say, two and a half times.
A special case is multiplication by -1, which leads to the concept of the additive inverse, and to subtraction, the inverse operation to addition.
The most general version of these ideas is the linear combination, where any number of terms are included in the generalised sum any number of times.
Useful sums
The following are useful identities:
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In general, the sum of the first n mth powers is
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where is the kth Bernoulli number.
The following are useful approximations (using theta notation):
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| for every real constant c greater than -1; |
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| for every real constant c greater than 1; |
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| for all nonnegative real constants c and d; |
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| for all nonnegative real constants b > 1, c, d. |
