can anyone give me a correct explanation for this problem..?

Question

Determine whether the sequence {a(n)} converges or not when
a(n)= 1 * 4 * 9.... (5n-1) / ((5n)^n)

i know it converges and has a limit of 0 but i dont know the explanation for this...

Answer 1

An easy way to show this is to note that for all n > 2,

n^n > n^2,

so

(5n - 1)/(5n)^n < (5n - 1)/(5n)^2

This sequence on the right can be written as

(5/n - 1/n^2) / 25,

and this obviously goes to zero as n becomes large.

Answer 2

a(n+1)/a(n) = (5n+4)/5(n+1) * 1/(1+1/n)^n ==> 1/e <1
so a(n) is monotone decreasing and bounded inferiorly by 0
So a(n) has a limit we call L
This proven
a(n+1)/a(n) = 1/e+b(n) with b(n) ==>0
a(n+1) = 1/e*a(n) +b(n) *(a(n)
As the limit exists we can write
L=1/e*L as b(n) *(an) ==>0
so L(1-1/e)=0 and so L = 0

Answer 3

www.math.binghamton.edu/calc2/PracticeEx2Fall05.pdf
www.math.binghamton.edu/calc2/Exam2_Solutions_Fall05.pd... www.math.binghamton.edu/calc2/Exam2_Solutions_Fall05.pdf
www.stevens-tech.edu/golem/ma116/Old_Exams/98_116Exam2

sure hope you can use one of these to help with your answer