Suppose

\[ \begin{aligned} a_n = f(n) \,\,\,\,\, \text{ where \(f\) is a decreasing and positive function.} \end{aligned} \]

If

\[ \begin{aligned} \int_1^\infty f(x)\,dx \,\,\,\, \text{ is finite,} \end{aligned} \]

then

\[ \begin{aligned} \int_1^\infty f(x)\,dx \,\,\, \le \,\,\, \sum_{n=1}^\infty a_n \,\,\, \le \,\,\, a_1 + \int_1^\infty f(x)\,dx \end{aligned} \]

!

You can find more information at Calculus Two: Sequences and Series.

# posted by rot13(Unafba Pune) @ 9:35 PM