On page 332 of Mark Ryan's book, Calculus for Dummies, he discusses a useful tip to try, before launching into an elaborate test to determine whether an infinite series diverges or converges. The tip is to just compare the numerator and the denominator of the expression in question; if the numerator is "larger" than the denominator, then the series diverges, otherwise, it converges.
And to make the tip quick and easy to use for purposes of comparison, when inspecting the numerator and denominator, he lists four expressions listing them from "smallest" to "biggest," as follows: n^10, 10^n, n!, and n^n.
This is where my question arises. When I plug in a number for n in those expressions, they don't fall in the same order as he states; for example, if I plug 5 into "n," then I get the following: n! = 5! = 120 ; n^n = 5^5 = 3,125 ; 10^n = 10^5 = 100,000, and ; n^10 = 5^10 = 9,765,625. In other words, my list, in ascending order, would be n!, n^n, 10^n, n^10, which is different than his list.
Am I missing something here? Am I not understanding what he means? Or is he wrong?
Any help would be much appreciated.
And to make the tip quick and easy to use for purposes of comparison, when inspecting the numerator and denominator, he lists four expressions listing them from "smallest" to "biggest," as follows: n^10, 10^n, n!, and n^n.
This is where my question arises. When I plug in a number for n in those expressions, they don't fall in the same order as he states; for example, if I plug 5 into "n," then I get the following: n! = 5! = 120 ; n^n = 5^5 = 3,125 ; 10^n = 10^5 = 100,000, and ; n^10 = 5^10 = 9,765,625. In other words, my list, in ascending order, would be n!, n^n, 10^n, n^10, which is different than his list.
Am I missing something here? Am I not understanding what he means? Or is he wrong?
Any help would be much appreciated.