Limits of Sequences

[johnjones]

I'm not sure how to tackle this question, where I'm asked to investigate the limit of the sequence: (1/2)^2, (2/3)^3, (3/4)^4, ... , (99/100)^100, ...
I don't know to obtain an answer (exact if possible, or with several decimal places). Thanks. I was told that this limit is an exaple of the form 1^infinity. Thanks.

 

[stapel]

What is the limit of the fractions, without the powers?

Note that the fraction "n/(n + 1)" can be restated as "(n + 1 - 1)/(n + 1) = (n + 1)/(n + 1) - 1/(n + 1) = 1 - 1/(n + 1)". What happens to 1/(n + 1) as n gets arbitrarily large?

Eliz.

 

[pka]

Problem is the equivalence, (1−[1/n])<SUP>n</SUP>.
It may surprise that the answer evolves a power of the number e.