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 nth 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