1. **State the problem:** We want to find $a$ and $\log_e L$ given the limit $$L = \lim_{n \to \infty} \frac{a \times \sum_{r=1}^m \frac{\tau^{b} r^n}{2n+1}}{n^6}$$ where $a \in \mathbb{R}$, $n \in \mathbb{N}$, and $L$ is nonzero and finite. 2. **Analyze the summation:** The sum is $$\sum_{r=1}^m \frac{\tau^{b} r^n}{2n+1} = \frac{\tau^{b}}{2n+1} \sum_{r=1}^m r^n$$ As $n \to \infty$, the term $r^n$ grows exponentially with $r$. The largest term dominates the sum. 3. **Dominant term in the sum:** The largest term is when $r = m$, so $$\sum_{r=1}^m r^n \sim m^n \quad \text{as } n \to \infty$$ Thus, $$\sum_{r=1}^m \frac{\tau^{b} r^n}{2n+1} \sim \frac{\tau^{b} m^n}{2n+1}$$ 4. **Rewrite the limit using dominant term:** $$L = \lim_{n \to \infty} \frac{a \times \frac{\tau^{b} m^n}{2n+1}}{n^6} = a \tau^{b} \lim_{n \to \infty} \frac{m^n}{(2n+1) n^6}$$ 5. **Behavior of the limit:** - If $m > 1$, then $m^n$ grows exponentially, so the limit diverges to infinity. - If $m = 1$, then $m^n = 1$, and the limit becomes $$L = a \tau^{b} \lim_{n \to \infty} \frac{1}{(2n+1) n^6} = 0$$ - If $m < 1$, then $m^n \to 0$, so $L = 0$. 6. **Condition for $L$ to be nonzero finite:** The only way for $L$ to be finite and nonzero is if the exponential growth is neutralized. Since $m^n$ dominates, this is only possible if $m=1$ and the numerator and denominator balance. 7. **Conclusion:** For $L$ to be nonzero finite, - $m=1$ - Then $$L = a \tau^{b} \lim_{n \to \infty} \frac{1}{(2n+1) n^6} = 0$$ which contradicts $L$ being nonzero. Therefore, the problem as stated implies no finite nonzero $L$ unless $a=0$ or other parameters change. **If we assume $m=1$ and redefine the problem to avoid zero limit, the only way is if the numerator grows like $n^6$ to cancel denominator. Since $m^n$ dominates, this is impossible unless $m=1$ and $a=0$ or $L=0$. Hence, the problem has no solution for nonzero finite $L$ unless $a=0$ and $L=0$. **Summary:** - $a$ must be 0 for $L$ finite and nonzero. - Then $L=0$. - $\log_e L$ is undefined for $L=0$. **Final answers:** $$a = 0$$ $$\log_e L \text{ is undefined (since } L=0)$$