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Arithmetic sequence diverge or converge formula?

My mind is drawing an absolute blank right now.

I recall some arithmetic and geometric formulas.

Arithmetic:
nth term Un=a+(n-1)d
Term to term Un+1=Un+d
Sn=n(a+l)/2

Is there a rule to find the limit for an arithmetic sequence? I understand that all arithmetic series diverge. I suppose it would be the same for its sequence, too.

I know some geometric don't converge and some do based on the |r|<1 converging. But you have a formula to limit. S∞=a/1-r
Reply 1
Original post by KingRich
My mind is drawing an absolute blank right now.
I recall some arithmetic and geometric formulas.
Arithmetic:
nth term Un=a+(n-1)d
Term to term Un+1=Un+d
Sn=n(a+l)/2
Is there a rule to find the limit for an arithmetic sequence? I understand that all arithmetic series diverge. I suppose it would be the same for its sequence, too.
I know some geometric don't converge and some do based on the |r|<1 converging. But you have a formula to limit. S∞=a/1-r

An arithmetic sequence is basically a straight line (well a set of samples lying on it) so not surprisingly it heads off to +/-inf. An arithmetic series is the corresponding sum (or ~integral) which is a quadratic (integrate a linear function to get a quadratic) so again it heads off to +/-inf. So neither converge as n->inf.
Original post by KingRich
My mind is drawing an absolute blank right now.
I recall some arithmetic and geometric formulas.
Arithmetic:
nth term Un=a+(n-1)d
Term to term Un+1=Un+d
Sn=n(a+l)/2
Is there a rule to find the limit for an arithmetic sequence? I understand that all arithmetic series diverge. I suppose it would be the same for its sequence, too.
I know some geometric don't converge and some do based on the |r|<1 converging. But you have a formula to limit. S∞=a/1-r

I think you are confusing APs with Geometric progressions which can converge if the common ratio, r <1
Reply 3
Original post by mqb2766
An arithmetic sequence is basically a straight line (well a set of samples lying on it) so not surprisingly it heads off to +/-inf. An arithmetic series is the corresponding sum (or ~integral) which is a quadratic (integrate a linear function to get a quadratic) so again it heads off to +/-inf. So neither converge as n->inf.

I see, that would explain why there’s no formula to find its limit but why do I recall something in my studies about applying a limit. Perhaps that was first principles.

An arithmetic series is a quadratic? Erm, Sn= n[2a+(n-1)d]/2 mmm, ax²+bx+c. I don’t see the connection lol
Reply 4
Original post by Muttley79
I think you are confusing APs with Geometric progressions which can converge if the common ratio, r <1

Yeah, I just wanted to be certain as it’s been a while since I’ve done series and sequences and when I first touched on the subject, it wasn’t all that clear the first time. I understand there’s something called oscillating sequence as well which follows the same rule as the geometric sequence?
Reply 5
Original post by KingRich
I see, that would explain why there’s no formula to find its limit but why do I recall something in my studies about applying a limit. Perhaps that was first principles.
An arithmetic series is a quadratic? Erm, Sn= n[2a+(n-1)d]/2 mmm, ax²+bx+c. I don’t see the connection lol

Theyre linear/quadratic in n. So the position to term formula is (assuming u_1 = a)
u_n = a + d(n-1) = dn + (a-d)
which is linear in n, and the series (sum to n) is
S_n = n/2(a + a+d(n-1)) = (d/2)*n^2 + n(a-d/2)
so its quadratic in n (which you should expect if you sum/integrate a linear seuence/function).

Obviously they dont converge as n->inf.

A geometric sequence can be oscillating if r=-1 for instance so
1,-1,1,-1,1,....
so a=1 and r=-1. It satisfies the usual common ratio test.
(edited 1 month ago)

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