Re: help with series!!

On Jul 8, 2006 11:21 PM CT, bill wrote:

If we assume the sum of all n (from n=1 to infinity)
an is convergent.
Let bn = sum of all ak's from k=n to infinity and
ak = an + an+1 + an+2 +
· · ·
how can we that lim (as n goes to infinity) bn = 0.
I missed a few classes and now i'm trying to figure
this out myself.
Can anyone give me a few hints at least?

There is a test called the "limit test" or the "n'th term
test" that says:

If lim_{n -> oo} a_n =! 0, then the sum_{n = 1}^{oo} a_n
is divergent.

In other words, if the general term does not run to zero
as n goes to infinity, then the infinite sum does not
converge.

It is sometimes useful to consider the contraposition of
this test, that is:

If the sum_{n = 1}^{oo} a_n converges, then the limit
lim_{n -> oo} a_n = 0.

This latter part may be helpful to you in showing that
b_n -> 0 as n -> oo.

Regards,
Kyle Czarnecki

P.S. Note that the converse of the contraposition is not
always true. That is, it need not be the case that:

If the limit lim_{n -> oo} a_n = 0, then the summation
sum_{n = 1}^{oo} a_n is convergent.

A classic example of this is the harmonic series.
.

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