Mathematics

Problem

Let

f

be a continuous, positive, decreasing function on

[1,\infty)

for which

\displaystyle\int_{1}^{\infty} f(x)\,dx

converges. Write an inequality that gives lower and upper bounds for the series

\displaystyle\sum_{n=1}^{\infty} f(n)

in terms of

f(1)

and the improper integral.