Problem finding the limit of the remainder of a series

Suppose there is a series

in which the last term comes from adding the remaining terms as a

geometric series with the first term

and ratio Integrating both sides of equation from

to gives

where

The denominator of integrand is greater than or equal to 1;hence

if , the right side of this inequality approaches zero as

Therefore

Now my question is how the book found the limit of as 0?

Book showed and I understand that the limit of is 0 when n

approaches . But that's only half of it? The author did not calculate

the limit of integral as a whole. I don't understand how he

reached the conclusion? Obviously I'm missing something simple. Is

there any theorem that i'm unaware of? Anyone knows how

he reached the conclusion?