Calculus BC: Series
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 .
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 n th 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,
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f (1)≥
f (x)dx
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and so on.
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
|
exists. If so, denote this limit by L . Then the value to which the sum converges satisfies the following inequality:
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L≤
f (n)≤f (1) + L
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