Brad Rodgers
Posted on Monday, 13 January, 2003 - 01:25 am:
Suppose we are given a sequence an such that


N
å
n=1 
 an ~ b N

,

for a constant b.

Does this necessarily imply that


N
å
n=1 
 an2 ~ c N

,

for a constant c?

Thanks,

Brad

David Loeffler
Posted on Monday, 13 January, 2003 - 09:31 am:
What if we set a2n=Ön and a2n+1=1-Ön? Then
N
å
n=1 
 an ~ 1
2
 N

, but
N
å
n=1 
 an2 ~ 1
6
 N3/2

or something like that.

Maybe the result is true if we constrain an to be positive real.

David

Michael Doré
Posted on Monday, 13 January, 2003 - 05:10 pm:
It's not. Consider a sequence where for each n either a2n=a2n+1=1/2 or a2n=3/4, a2n+1=1/4. Then
N
å
n=1 
 an ~ N/2

but it should be clear that we make
N
å
n=1 
 an2/N

oscillate between 1/4 and 5/16.
Brad Rodgers
Posted on Tuesday, 14 January, 2003 - 02:26 am:
Hrm. I vaguely recall seeing this argument before, or at any rate something very close to it - Michael's that is; in fact, I vaguely recall asking something to the effect of this question before...

Apologies if those memories are accurate, but thanks in either case for the quick responses.

Brad