Thread: Convergence- proving bounded

Given sequence, a= 1/1^2+1/2^2+1/3^2+.......+1/n^2

(1) I have to prove for natural numbers n, such that n is greater than or equal to 2,

a is less than or equal to 1+1/1*2+1/2*3+1/3*4+.....+1/(n-1)*n

(2) Prove that a is bounded for natural numbers

I've been given a hint in the question to use the fact that for n greater than or equal to 2, 1/(n-1)n=1/(n-1) -1/n

Anyone have any ideas? Anything would be very much appreciated

Hello.

In part one, you can just compare terms directly. The -term in your first series is , and the -term in your second series is . Is it true that (for )? If so, then you have , and therefore, the first series is bounded by the second.

In part two, we can try to sum up the second series to put a bound on the first. Try writing it down using your hint:



Anything nice happen? What happens when you take ? Use this to conclude.