Changing Variables In Summation

Question

Isn't the change in the summation variables as intended by the author(Jeff Erickson) wrongly done??

As far as I understood that change in summation variables limits is alike the way we do in Integral Calculus..

If i is $0$ then j should be $ \lg n $

And

if i is $ \lg n $ then j should be $1$..

Could someone just explain me how is the variable being interchanged if I have

misunderstood the concept??

Thanks in Advance!!(Sorry for bad formatting)

Answer 1

I think the limit should be from $j=\log n$ to $j=1$.