Thread: Show 1/x+1 less than or equal to ln(x+1)/x

1/x+1 less than or equal to ln(x+1)/x
using definition of ln(x) as intergral of 1 to x of 1/t dt

show that for every real number we have x such that

Originally Posted by sbacon1991

1/x+1 less than or equal to ln(x+1)/x
using definition of ln(x) as intergral of 1 to x of 1/t dt
show that for every real number we have x such that

By definition: if then .

So which means we can use the mean value theorem.

If then

But implies

That gives .

Let and we have it.