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) + ^{ ... } |
Considering the following figure, we see that
f (2)≤ f (x)dx |
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 |
and so on. Thus we have
f (1) + f (2) + ^{ ... } + f (n)≤f (1) + f (x)dx |
But the left side of this inequality is simply the n th partial sum for the series under consideration. If
f (x)dx |
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 |
and so on.
Thus
f (1) + f (2) + ^{ ... } + f (n)≥ f (x)dx |
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:
L≤ f (n)≤f (1) + L |