Question

Proving. Help. :|

Answer 1

Arithmetic sequences do not converge

Answer 2

{0,0,0,0...}

Answer 3

@Zarkon what does that mean?

Answer 4

that is an arithmetic sequence that converges

Answer 5

@Zarkon :(( I need to prove that arithmetic sequences diverge :( How will that help in the proof?

Answer 6

The nth partial sum is given by:\[S_n=n(\frac{a_1+a_n}{2})\]Now the limit as n approaches infinity does not approach a fixed number. If you count {0,0,0,0...} as an arithmetic sequence then yeah, this one converges.

Answer 7

you need to add a condition to your question because it is not true. (like non-constant arithmetic sequences do not converge)

Answer 8

Wait. We have discussed a while ago that there exists a sequence called constant sequence. {0,0,0,...} falls under it. :) @Zarkon :)

Answer 9

any sequence of the form \(a_n=a\) is an arithmetic sequence that converges to \(a\)

Answer 10

I just realized that I was answering the wrong question. You wanted the sequence, not the series.

Answer 11

@eseidl Yes :)

Answer 12

considering {0,0,0,..} is not an arithmetic sequence, how am I going to prove that?

Answer 13

but it is.

Answer 14

\[a_n=a_1+(n-1)d\]If\[d \neq0\]than the limit as n approaches infinity doesn't exist and therefore any non-constant arithmetic sequence is non-convergent. You'll notice this equation describes a straight line, which doesn't approach a fixed value unless d=0. But keep in mind that it isn't really a straight line but a collection of discrete points that would lie on a line.

Answer 15

if you have \(a_n=a+b(n-1)\)

then if \(b\ne 0\) then \(a_n\to \infty\) if \(b>0\) and \(a_n\to -\infty\) if \(b<0\)

Answer 16

THANK YOU @Zarkon & @eseidl :) BIG THANKS!