### Home > CALC > Chapter 4 > Lesson 4.2.1 > Problem4-50

4-50.

If $n$ is a positive integer write an integral to represent $\lim\limits_ { n \rightarrow \infty } \frac { 1 } { n } [ \frac { 1 } { ( \frac { 1 } { n } ) } + \frac { 1 } { ( \frac { 2 } { n } ) } + \ldots + \frac { 1 } { ( \frac { n } { n } ) } ]$.

Notice that this is a Riemann sum with infinitely many rectangles.

And a Riemann sum with infinitely many rectangles is the Definition of an Integral:
Can you rewrite this as an integral?

$\lim\limits_{n\rightarrow \infty }\frac{1}{n}=\lim\limits_{x\rightarrow 0}\Delta x=dx..............$ so we can substitute $\frac{1}{n}$ with $dx$.

The $dx$ represents the infinitely small width of each rectangle.
Now let’s find the height of each rectangle.
Heights, of course, are represented by a function, $f(x)$.
But what is $f(x)$?

Since $x$ is a variable, we will let $x$ represent the part of the series that is changing:
This is beginning to look more like an integral: $\lim\limits_{n\rightarrow \infty }\frac{1}{x}dx$

We still need to find the bounds of the integral.
The lowest value of $x$ is $\frac{1}{n}$  . Since $\lim\limits_{n\rightarrow \infty }\frac{1}{n}=0$, the lower bound is $0$.