Convergent limits for sequences: picture terms

[939]

A limit of a sequence is definitely convergent if:

If for any value of K there is an N sufficiently large that an > K for n > N, OR for any value of K there is an N sufficiently large that an<±K for n > N

My only question is what exactly are K, N, an and n? What values are they? How would they be graphed? I.e. for the sequence a(n) = 2n

n = 2
an = 4
What are K and N? Are they on the horizontal or vertical axis?

 

[tiny-tim]

hi 939!

A limit of a sequence is definitely convergent if:
…
My only question is what exactly are K, N, an and n? What values are they? How would they be graphed?

nooo, you mean definitely divergent

essentially, it means that the sequence converges to ∞, or to -∞

we can't use δ and ε for ∞ (because we can't get close enough to ∞ !)

so instead of small circles round the limit, we draw large circles round the limit (∞), ie x > K (or x < -K), and we say that that large circle has to contain all a_n once n is large enough

it's the same as saying that the sequence {1/a_n} has the limit 0 (from above, for the ∞ case) (or from below, for the -∞ case)​