Question

i have two limits and infinite series questions:

(1) Given B find M such that when n > M then (n^n)/n! > B

(2) Given € > 0 find M such that when n > M then (n^n)/n! < €

Answer 1

your first question is correct, but let me tell you it is impossible to n>M always so that your 2nd hold, e.g if you choose €=1/e then you never get such a natural number which satisfies your inequality.you can find upper bound of n for this to hold which will obviously depend upon € but not any lower bound M of n.if you like this answer please rate it.

note that i have given a bound for n which can be found from wikipedia article about stirllings approximation.