Show that $\lim_{n\to\infty}\sqrt[\sum_\limits{k=0}^{n}\lambda_k]{\prod_\limits{k=0}^{n}a_k^{\lambda_k} }= \ell =\lim_{n\to\infty} a_n$
Solution 1:
The question is equivalent to:$$\lim_{n\to\infty}\frac{{\sum\limits_{k = 0}^n {{\lambda _k}\ln ({a_k})} }}{{\sum\limits_{k = 0}^n {{\lambda _k}} }} = \ln l$$ Since ${\sum\limits_{k = 0}^n {{\lambda _k}} }$ increases to $+\infty$, Stolz's theorem immediately gives the reult.