Question

How do you find the Limit of `n^( ln( (n+1)/n )` as n approaches infinity?

Answer 1

1

Explanation:

Use `(ln n)/n^k to 0`, as`n to oo, k = 1, 2, 3, ..`

Let `f = n^(ln((n+1)/n)=n^(ln(1+1/n))`.

`log f = ln(1+1/n)ln n`

`=ln n(1/n-1/(2n^2)+1/(3 n^3)..`

`to 0-0+0-...,`, (alternately `0 and -0)` as `n to oo`.

So, `ln f to 0 and f to1`.. .