Animated Solution for Mathematics - Definite Integration: ${\lim }_{n\to \infty }\frac{1^p+2^p+3^p+\cdots +n^p}{n^{p+1}}$

The expression given is ${\lim }_{n\to \infty }\frac{1^p+2^p+3^p+\cdots +n^p}{n^{p+1}}$. At first glance, this appears to be a mountain of complexity involving an infinite sum of terms raised to the power of $p$.

In the world of JEE Advanced, such complexity is often a mask for hidden simplicity. We shall avoid brute-force methods and instead utilize the power of visualization and calculus.

First, we tame the numerator by expressing it in the compact language of summation: ${\sum }_{r=1}^n{r}^p$.

Next, we address the denominator $n^{p+1}$. By the laws of exponents, we know that $n^{p+1}=n^p\cdot n$. This split is crucial because it allows us to pair the $n^p$ with the $r^p$ in the numerator.

We rewrite the expression as follows:

We rearrange the terms to isolate the components of a Riemann sum. By pulling the $1/n$ outside the $r^p$ term, we obtain:

This matches the standard form of a Riemann sum, defined as ${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}f\left(\frac{r}{n}\right)$. Here, our function is $f(x)=x^p$, where $1/n$ represents the width of the rectangles ($\Delta x$) and $r/n$ represents the position $x$.

To transition from the discrete to the continuous, we define our boundaries. As $n\to \infty$, the lower limit $r=1$ becomes $1/n\to 0$, and the upper limit $r=n$ becomes $n/n=1$.

Consequently, the infinite sum morphs into the following definite integral:

Integrating $x^p$ is a fundamental application of the power rule. We add one to the exponent and divide by the new exponent:

Applying the limits from $0$ to $1$, we calculate:

Since $1$ raised to any power is $1$, the expression simplifies to the final result:

We have successfully transformed a terrifying limit into a simple geometric area. Whenever you encounter a limit of a sum, look for the $1/n$ and the $r/n$ structure; the integral is always waiting to be revealed.