Find the following limit:
$$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$$
I had an idea of using upper Riemann sum for function $x^4$ on interval $[0,1]$ but I don't know how to deal with $k^5$ in denominator which ruins my approach. Kindly asking for some help.
Find limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$.
calculusintegrationlimitsriemann sumsummation
Best Answer
Using 'Definite integrals as limit of a sum':
The above limit is equivalent to $$\int_0^1{\frac{x^4}{x^5+1}}dx.$$
Substituting $z=x^5$ gives us $$\int_0^1{\frac{1}{5(z+1)}}dz$$ which is equal to $\displaystyle\frac{\log{2}}{5}$.