ANSWER
[tex]\lim _{n\to\infty}\text{ }(\frac{3n^5^{}}{6n^6+1})\text{ = 0}[/tex]
EXPLANATION
Step 1: Given that:
[tex]\sum ^{\infty}_{n\mathop=1}(\frac{3n^5^{}}{6n^6+1})[/tex]
Step 2: Expand the limit
[tex]\begin{gathered} \lim _{n\to\infty}(\frac{3n^5^{}}{6n^6+1})\text{ } \\ \text{ = }\lim _{n\to\infty}\frac{n^5}{n^5}(\frac{3^{}}{6n^{}+\frac{1}{n^5}}) \\ \text{ = }\lim _{n\to\infty}(\frac{3^{}}{6n^{}+\frac{1}{n^5}}) \\ =\text{ }\lim _{n\to\infty}(\frac{3^{}}{6(\infty)^{}+\frac{1}{(\infty)^5}}) \\ \text{ = }\lim _{n\to\infty}(\frac{3^{}}{6(\infty)^{}+\frac{1}{\infty}}) \\ \text{ = }\lim _{n\to\infty}(\frac{3^{}}{6(\infty)^{}+0^{}}) \\ \text{ = }\lim _{n\to\infty}(\frac{3^{}}{6(\infty)^{}^{}})\text{ = 0} \\ \end{gathered}[/tex]
Hence,
[tex]\lim _{n\to\infty}(\frac{3n^5}{6n^6+1})\text{ = 0}[/tex]
