What is \(\sum _{r=1}^n r^7 \)?

Calculus Level 4

Let \( \displaystyle M_n= \sum_{r=1}^n r^4 , N_n= \sum_{r=1}^n r^2 \) and \(\displaystyle Y_n= \sum_{r=1}^n r^7 \). If \(\displaystyle \lim_{n\to\infty} \dfrac{M_n N_n}{Y_n} \) can be expressed as \( \dfrac AB\), where \(A\) and \(B\) are coprime positive integers.

Find \(A+B\).


You may want to read the wiki page: Method of undetermined coefficients.
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