Animated Solution for Mathematics - Definite Integration: ${\lim }_{n\to \infty }\frac{1^4+2^4+3^4+\cdots +n^4}{n^5}-{\lim }_{n\to \infty }\frac{1^3+2^3+3^3+\cdots +n^3}{n^5}$

We are tasked with evaluating the limit:

Instead of using tedious summation formulas, we utilize the Riemann Sum identity, which acts as a bridge between discrete sums and continuous integration:

Let the first term be $L_1={\lim }_{n\to \infty }\frac{{\sum }_{r=1}^n{r}^4}{n^5}$. To apply the identity, we rewrite the expression to isolate the $\frac{1}{n}$ factor:

By substituting $\frac{r}{n}=x$ and $\frac{1}{n}=dx$, the summation transforms into a definite integral:

Evaluating this integral yields:

Now, consider the second term $L_2={\lim }_{n\to \infty }\frac{{\sum }_{r=1}^n{r}^3}{n^5}$. We manipulate the denominator to reveal the Riemann structure:

The bracketed expression $\frac{1}{n}{\sum }_{r=1}^n(\frac{r}{n}{)}^3$ converges to ${\int }_0^1{x}^{3}dx=\frac{1}{4}$. However, this is multiplied by an additional factor of $\frac{1}{n}$:

The higher power in the denominator effectively causes the sum of cubes to vanish in the limit. Combining our results, we find:

This approach demonstrates that understanding the growth rates and the geometry of Riemann sums is far more efficient than algebraic expansion.