Re: Rational vs irrational

On 22 Sep 2005 00:24:07 -0700, "statcat" <lusots@xxxxxxxxx> wrote:

>it is not series, but sequence.
>Can anyone give me an example for a sequence of rational numbers having
>a limit that is an irrational number?
>
>I have one for the opposite: a sequence of irrational number having a
>limit that is rational number: i.e. for (n=1, 2, .....) the sequence
>1/sqroot(n) will have a limit of 0
>if n -> OO

Given any series a1+a2+a3+... of rational terms converging to an
irrational number a, the sequence of partial sums s1,s2,s3, ... is a
sequence of rational numbers with the same limit a. In other words,
let s1,s2,s3, ... be given by:

s1=a1
s2=a1+a2
s3=a1+a2+a3
...

Then since the terms of the series are rational, so are the partial
sums, and by the definition of convergence of a series, if the series
a1+a2+a3+... converges to a, then the sequence s1,s2,s3,... also
converges to a.

quasi
.

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