The Integral Test
Suppose we have a function f (x), defined for all x≥1, which is positive and
decreasing. This function defines a sequence {f (n)} and a series
 f (n) = f (1) + f (2) + ... |
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Considering the following figure, we see that
f (2)≤ f (x)dx |
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since a rectangle with height f (1) and width 1 is contained within the region below the
graph of f from 0 to 1.
Figure 4.1: The Function Contains the Rectangular Regions
Similarly,
f (3)≤ f (x)dx |
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and so on. Thus we have
f (1) + f (2) + ... + f (n)≤f (1) +  f (x)dx |
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But the left side of this inequality is simply the nth partial sum for the series under
consideration. If
 f (x)dx |
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is defined, then the partial sums are bounded, so the series converges.
This logic goes the other way too. As the figure below demonstrates,
| f (1)≥f (x)dx |
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and so on.
Figure 4.2: The Rectangular Regions Contain the Function
Thus
f (1) + f (2) + ... + f (n)≥ f (x)dx |
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If
does not exist, then the integral becomes
arbitrarily large for large
n, as do the partial sums for the series. Therefore the series
f (n) does not converge. We summarize the results of this section in
the following statement.
If
f is a positive, decreasing function defined for
x≥1, then
f (n) converges if and only if the limit
| f (x)dx |
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exists. If so, denote this limit by L. Then the value to which the sum converges
satisfies the following inequality:
| L≤f (n)≤f (1) + L |
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