# Homework Help: Convergence of a sequence

1. Sep 20, 2015

### andyfeynman

1. The problem statement, all variables and given/known data
Show that the sequence {xn}:
xn := (21/1 - 1)2 + (21/2 - 1)2 + ... + (21/n - 1)2 is convergent.

2. Relevant equations

3. The attempt at a solution
If n > m,
|xn - xm| = (21/n - 1)2 + (21/(n-1) - 1)2 + ... + (21/(m+1) - 1)2
< (21/n)2 + (21/(n-1))2 + ... + (21/(m+1))2
< (21/(m+1))2 + (21/(m+2))2 + ...
= 41/m
Let ɛ > 0. We choose N such that 41/N < ɛ for all n > m > N.
Then |xn - xm| < 41/N < ɛ for all n > m > N.

2. Sep 20, 2015

### Staff: Mentor

Which N would you choose for ɛ=1?

3. Sep 20, 2015

### andyfeynman

Just forgot it. I made a very stupid mistake.
But I came up with the idea of letting bk = 21/k - 1.
This means bk2 < 4[k(k-1)] for all k > 2.
Therefore,
xn = 1 + b22 + ... + bn2
< 1 + 4/[1(2-1)] + ... 4/[n(n-1)]
= 1 + 4(1 - 1/2) + ... + 4[1/(n-1) - 1/n]
= 5 - 4/n
< 5
Since xn is monotone increasing and bounded, it is convergent.
But is there any way to do it using the Cauchy criterion?

4. Sep 20, 2015

### Staff: Mentor

That step is certainly not trivial.

I would use something like 21/k < 1 + c/k for some c.

Cauchy criterion: Probably, but I don't see how it would help.