Show this sequence converges to zero?

Question

lim: n^(1/2)*(1-ln((1+1/n)^n))

How do I show this sequence converges to zero? It would be so easy if that n^(1/2) wasn't there since ln((1+1/n)^n)->ln(e)=1

But I'm stuck and I don't know what to do.

Maybe it's easier if you write it as:
n^(1/2) *(1- n*ln(n+1) +n*ln(n))

Any help would be greatly appreciated

Answer 1

Look at the function f(x) = x^(1/2)*[1 - ln(1+ 1/x)^x], and write it as:
f(x) = x^(1/2)*[1 - xln(1 + 1/x)] = [1 - xln(1 + 1/x)]/(1/x^(1/2))

Now apply L'Hospital:

[-ln(1 + 1/x) + 1/(1+x)]/[-(1/2)/x^(3/2)] --> 0...when x --> oo

Regards
Tonio

Answer 2

Remember: the limit of a product is the product of the limits, if both limits exist and are finite. If you are taking the limit as x -> 0, then lim(n^(1/2)) = 0. If you are taking the limit as n -> inf, you have the indeterminate form 0*inf. Write the function in the form [1 - ln((1 + 1/n)^n]/[1/n^(1/2)] and use L'Hospital's Rule.

Answer 3

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