Animated Solution for Mathematics - Definite Integration: Show that: ${\lim }_{n\to \infty }\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{6n}\right)=\log 6$.

We are tasked with evaluating the limit as $n\to \infty$ of the sum:

At first glance, this appears to be a chaotic, never-ending list of fractions. However, in mathematics, we seek order within chaos. When you encounter a sum of $n$ terms where $n$ grows to infinity, your intuition should immediately point toward a Riemann Sum.

To decode the pattern, we observe the denominators $n+1,n+2,\dots ,6n$. We can express the last term, $6n$, as $n+5n$.

This reveals that the variable part, which we denote as $r$, ranges from $1$ to $5n$. We can now represent the entire expression using Sigma notation:

To convert this limit into a definite integral, it must strictly adhere to the standard Riemann form:

We require a factor of $\frac{1}{n}$ outside the summation, and the function inside must depend solely on the ratio $\frac{r}{n}$. We achieve this by factoring $n$ out of the denominator:

Substituting this back into our series, we obtain:

This expression serves as the bridge between discrete algebra and continuous calculus. As $n\to \infty$, the width of each rectangle $\frac{1}{n}$ becomes the infinitesimal $dx$, the ratio $\frac{r}{n}$ becomes the continuous variable $x$, and the summation sign transforms into an integral.

The function we are integrating is $f(x)=\frac{1}{1+x}$. To determine the limits of integration, we evaluate the bounds of the ratio $\frac{r}{n}$ as $n\to \infty$.

The lower limit $a$ is:

The upper limit $b$ is:

We have successfully translated the infinite series into a clean, elegant definite integral representing the area under the curve from $x=0$ to $x=5$:

Executing the integration, we find:

Applying the limits of integration:

Since $\ln (1)=0$, the final result is: