Re: Infinite series question

jane <jane1806@xxxxxxxxxx> writes:

jane <jane1806@xxxxxxxxxx> writes:

Suppose, x_i in R, 0 < x_i < 1, sum_i x_i = oo,
and

A_n = sum_{i <= i <= n} x_i, so that A_n -> oo.

B_n = sum_{1 <= i <= n} x_i exp[-A_n * xi].

Are there any known results under which
conditions on
(x_i), we have that B_n -> 0 as n -> oo ?

I forgot to mention in the previous post, that it
is also known that lim
x_i = 0 and i -> oo.


It's true if x_i > c i^(-p) for constants c > 0 and 0
< p < 1/2.
A_n > c int_1^(n+1) t^(-p) dt = c/(1-p) ((n+1)^(1-p)
- 1) > b n^(1-p)
for n sufficiently large, if 0 < b < c/(1-p).

Since ln(t)/t is decreasing for large t, for n
sufficiently large we have
2 ln(A_n)/A_n < 2 (ln(b) + (1-p) ln(n))/(b n^(1-p)) <
c n^(-p) < x_i for
1 <= i <= n.

Then exp(-A_n x_i) <= exp(-2 ln(A_n)) = A_n^(-2) so
B_n <= sum_{i=1}^n x_i A_n^(-2) = A_n^(-1) -> 0 as n
-> infinity.

On the other hand, for x_i = i^(-1/2) we have
0 < A_n x_i <= 2 sqrt(2) for i >= n/2, and then
B_n >= A_n exp(-2 sqrt(2)) -> infty.
--

This is helpful anyway, but quite not but i was initially looking for.
I was looking for some good convergence of the sum

C_n = sum_{1<=i<=n} x_i * prod_{1<=j<=n}(1-x_i * x_j).

I just used the inequality 1-t <= exp(-t) and then hoping for already good
convergence of the sum

B_n = sum_{1 <= i <= n} x_i exp[-A_n * xi] posted the question above, but
that didn't help according to you answer.

Do you think, is it possible to give some weaker conditions on x_i so that
for new C_n defined above we can get that lim_n C_n under the same
conditions
0 < x_i < 1, x_i -> 0, sum x_i = oo ?

Not much weaker.

If 0 < t < a < 1, exp(-ct) <= 1 - t where exp(-ca) = 1 - a
(i.e. c = -ln(1-a)/a), so if all x_i <= sqrt(a),
C_n >= sum_{i=1}^n x_i exp(-c A_n x_i)
Then again x_i = b i^(-1/2) (where 0 < b < 1) would have
A_n x_i <= some positive constant for i >= n/2, and
C_n >= k A_n (for some positive constant k)
-> infinity as n -> infinity.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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