does the sum of a sequence of rational numbers produce a rational number, or not a rational number?

Question

anonymous · Answer

I'm pretty sure it's rational, but I'm not sure how I'd prove it. Maybe by induction?

The sum of the sequence $\left\{\dfrac{1}{2},\dfrac{1}{4}\right\}$ is rational, since $\dfrac{1}{2}+\dfrac{1}{4}=\dfrac{3}{4}$. So if you keep adding rational terms I don't see how it could suddenly become irrational.

amistre64 · Answer

i was thinking more along the lines of the zeta function,  but then myquesiton is prolly more about the limit of the sum as opposed to a definite value

amistre64 · Answer

$$\Large \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$

anonymous · Answer

A sequence of rational numbers does not necessarily have to add to give a rational number. A finite sum of rationals will always add to give a rational of course, which can be proved by induction as suggested. But sequences are by definition infinite.

The sequence $\frac{1}{n^2}$ for instance, sums to give $\frac{\pi^2}{6}$. (Not sure I can show that to be honest....)

amistre64 · Answer

euler did a nice job of showing it :)

anonymous · Answer

Isn't $s$ a complex number?

anonymous · Answer

Strictly speaking, a real number is a complex number...

amistre64 · Answer

s can be defined as a complex number, but can be rational

amistre64 · Answer

1/n^2 sums to pi^2/6  as Erin mentions

amistre64 · Answer

http://www.youtube.com/watch?v=wcjknbmTYOI this gives a nice rundown of the "showing"

amistre64 · Answer

around 35 to 37 gets into the meat of it .... unless you want to watch it from the beginning :)

anonymous · Answer

Oooh great :) We were told we'd see a proof of it when we study Fourier series next year, but an elementary proof would be nice.

amistre64 · Answer

i keep trying to learn about foureir series and keep falling asleep

amistre64 · Answer

having a pretty teacher might hold my interest, but then i run the risk of being distracted too much to learn :)

anonymous · Answer

With that introduction, I'll look forward to Fourier series next term then ;-) Just found this proof: http://www.maths.usyd.edu.au/u/daners/publ/abstracts/zeta2/zeta2.pdf Rather neat!