Suppose one has the sequences ${\displaystyle a_{n}={\frac {\ln ^{n}(b)}{n!}}}$. how to prove elegantly that series ${\displaystyle \sum _{n=0}^{\infty }a_{n}}$converges to b WITHOUT arguing from Taylor series? The expression of partial sum S_n is not extremely simple. Thank you. ---- — Preceding unsigned comment added by 95.51.14.27 (talk) 02:14, 16 January 2012 (UTC)[reply]

Ratio Test. --COVIZAPIBETEFOKY (talk) 04:35, 16 January 2012 (UTC)[reply]

That would prove it converges, not what it converges to. -- Meni Rosenfeld (talk) 09:11, 16 January 2012 (UTC)[reply]

You're not very clear about what we are allowed to use. In particular, usually either exp or ln is defined some way, and then the other is defined as its inverse. The proof you want may depend on your definitions. But once it is established that they are inverses, there's no avoiding the fact that ${\displaystyle \sum _{n=0}^{\infty }{\frac {\ln(b)^{n}}{n!}}=b}$is equivalent to ${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=\exp(x)}$, so you're basically asking to prove the Taylor series of exp.

And to do that, if you already know that ${\displaystyle \exp(x)=\lim _{n\to \infty }\left(1+{\tfrac {x}{n}}\right)^{n}}$, you can use the binomial expansion. -- Meni Rosenfeld (talk) 09:11, 16 January 2012 (UTC)[reply]