04 May 10
2 edits
Originally posted by adam warlockfair enough, wasn't actually trying to make a precise mathematical statement...just overly hastily highlighting one of the arguments that had me splitting my sides with laughter (and a + should have preceded the 5, not a - ) 🙂
That series doesn't converge.
04 May 10
Originally posted by AgergI saw the humr in your post and replied with mock seriousness. But I forgot to put the smiley... 😉
fair enough, wasn't actually trying to make a precise mathematical statement...just overly hastily highlighting one of the arguments that had me splitting my sides with laughter (and a + should have preceded the 5, not a - ) 🙂
05 May 10
2 edits
Originally posted by joe shmoI would say it has converged to pi.
the series still doesnt converge...correct?
Definition: "A series S{a(n)} is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit."
In this series, (S), it starts with 1+2-3+4+5-9+pi+10+11-21.
It has only one value at a(1) (a very short series), which happens to be pi, which is finite.
We can examine it by showing that sup(S) = inf(S) = pi, therefore S converges to pi.
It converges alright.
05 May 10
Originally posted by FabianFnasThe series does not converge! It bounces around rather wildly as its terms become more numerous.
I would say it has converged to pi.
Definition: "A series S{a(n)} is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit."
In this series, (S), it starts with 1+2-3+4+5-9+pi+10+11-21.
It has only one value at a(1) (a very short series), which happens to be pi, which is finite.
We can examine it by showing that sup(S) = inf(S) = pi, therefore S converges to pi.
It converges alright.
05 May 10
Originally posted by amolv06Let's see if I can explain.
The series does not converge! It bounces around rather wildly as its terms become more numerous.
(A): Do you think 3 + 5 is a series or a sum?
(B): Do you think {3, 5} is a series or a sum?
And (C): Do you think 1+2-3+4+5-9+pi+10+11-21 is a series or a sum?
And what about pi itself, does that value diverge or converge?
05 May 10
6 edits
Let's see if I can explain.
(A): Do you think 3 + 5 is a series or a sum?
(B): Do you think {3, 5} is a series or a sum?
And (C): Do you think 1+2-3+4+5-9+pi+10+11-21 is a series or a sum?
And what about pi itself, does that value diverge or converge?
A: Both
B: Neither.
C: Both
Pi is a number. The term convergence is generally applied to an infinite series or sequence. I think the question, though semantically correct, is meaningless. I could be wrong about this, though.
But this is largely irrelevant. We are asking whether the infinite series mentioned above converges. By your own definition this series does not converge. Your definition stated:
"A series S{a(n)} is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit."
No such finite limit exists for these partial sums. Or, if it does, as N approaches infinity, can you tell me this limit. Or can you prove that one exists? You can not.
S1 = 1
S2 = 3
S3 = 0
S4 = 4
S5 = 9
S6 = 0
S7 = pi
S8 = Pi+10
S9 = pi + 21
S10 = pi
S11 = pi + 22
S12 = pi + 45
S13 = pi
S14 = pi + 46
S15 = pi + 93
S16 = pi
This sequence continues like this forever of course. As N approaches infinity there is no limit that the sequence of partial sums converges to. Hell, the sequence isn't even bounded. One of the conditions for convergence is that the sequence must be bounded.
05 May 10
Originally posted by amolv06I see only one item in that series and that is the expression 1+2-3+4+5-9+pi+10+11-21. Note there are no commas in between, nor semicolon. If you evaluate this you get pi. A value.Let's see if I can explain.
(A): Do you think 3 + 5 is a series or a sum?
(B): Do you think {3, 5} is a series or a sum?
And (C): Do you think 1+2-3+4+5-9+pi+10+11-21 is a series or a sum?
And what about pi itself, does that value diverge or converge?
A: Both
B: Neither.
C: Both
Pi is a number. The term convergence is g ...[text shortened]... en bounded. One of the conditions for convergence is that the sequence must be bounded.
So this is a finite series of only one term. S1 = pi. There is no S2, nor S3, etc. Only S1.
What we discuss here is definitions and how we interprete the "1+2-3+4+5-9+pi+10+11-21". I know you know about series, and I think you know that I know some too. We can continue, but it doesn't lead to any meaningful discussion. So I stop here. You can have the last comment if you want.
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05 May 10
Originally posted by FabianFnasjust for clarification when I stated the series doesn't converge I thought of it in the way almov did, but because ( as fabian pointed out ) it is a series of only one term im going with fabian, if terms were separated by commas It doesn't converge.
I see only one item in that series and that is the expression 1+2-3+4+5-9+pi+10+11-21. Note there are no commas in between, nor semicolon. If you evaluate this you get pi. A value.
So this is a finite series of only one term. S1 = pi. There is no S2, nor S3, etc. Only S1.
What we discuss here is definitions and how we interprete the "1+2-3+4+5-9+pi+ ...[text shortened]... lead to any meaningful discussion. So I stop here. You can have the last comment if you want.
On a side note: It does leave me confused? Does the sum continue in a non-distinct way?
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06 May 10
Originally posted by joe shmoand again sorry, I see the how the pattern is unique now.😞
just for clarification when I stated the series doesn't converge I thought of it in the way almov did, but because ( as fabian pointed out ) it is a series of only one term im going with fabian, if terms were separated by commas It doesn't converge.
On a side note: It does leave me confused? Does the sum continue in a non-distinct way?