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