Another Series
 

When someone posts a series and asks what number comes next, in theory there are an infinite number of answers. All you need to do is to come up with a polynomial of degree n+1 where n is the number of terms in the given series. The first n solutions to the polynomial matching the first n terms in the series. You can then have another solution to the polynomial that is any value you desire.

Of course, when someone posts a series, there is the implied requirement that the rule be the simplest one possible and a high degree polynomial is not very simple.

With that in mind, what digits come next? (And why?)

3.141592... :question:

I think the next numbers must be
3,141592[size=18][b][color=red]653589793238462643383279502[/color][/b][/size]
because they are the digits of pi!!

greetz andi314 :D

[quote="andi314"]I think the next numbers must be
3,141592[size=18][b][color=red]653589793238462643383279502[/color][/b][/size]
because they are the digits of pi!!

greetz andi314 :D[/quote]

The problem is that pi is the limiting result of an infinite series. That is not very simple.

The simplest solution is 355/113

So the series really continues:

3.141592[color=red][b]920353982300884955752212[/b][/color]...

Another Series
 

That all depends on your metric for simplicity.

Your expression is a quotient that requires 7 symbols.

On many systems, I can express the other number with only one symbol.

[quote="Gary Edstrom"]The problem is that pi is the limiting result of an infinite series. That is not very simple.[/quote]
But limit-of-an-infinite-series is not the "simplest" definition of pi in most contexts.

Pi has far more fundamental significance, and appears much more often in an immense variety of contexts, than 355/113.

[quote]When someone posts a series and asks what number comes next, in theory there are an infinite number of answers.[/quote]
True, and when I was younger I'd reflexively spout out that one, as did my buddies.

But that's really a lazy answer which dodges sincere effort at using one's intelligence to determine the most logical or simplest continuation within the context of the problem.

[quote]All you need to do is to come up with a polynomial of degree n+1 where n is the number of terms in the given series.[/quote]
Yes, a cubic polynomial fitted to just 2 terms is quite flexible!

Once one has learned about fitting polynomials to given points, one can trot out this answer automatically (or for humorous intent) in response to "continue the series" problems -- as my friends and I did when we were young -- but that doesn't make it the most intelligent answer in most contexts.

I'm definitely out of my league if people were doing these things when they were children and they always completely elude me.

Oh, don't be fooled by my photo! It's not recent! ;)

When I wrote "young" I meant "young adult".

[quote="Gary Edstrom"][quote="andi314"]I think the next numbers must be
3,141592[size=18][b][color=red]653589793238462643383279502[/color][/b][/size]
because they are the digits of pi!!

greetz andi314 :D[/quote]

The problem is that pi is the limiting result of an infinite series. That is not very simple.

The simplest solution is 355/113

So the series really continues:

3.141592[color=red][b]920353982300884955752212[/b][/color]...[/quote]


Ah, by "simplest" you mean the rational fraction with the smallest denomimator. Fair enough. There is at least a simple algorithm for determining it: the continued fraction expansion.

Paul