Finding equations of summations from 1 to a non integer

[Gouskin]

Firstly, I'm in prealculus, so I don't really know any of the notation
Is there a theorem to find the equation of a summation that would allow me to sum the terms from, say, 1 to 2.6?

For example:

And if x is all real numbers, the equation in summation notation won't return an answer for any non integers, while the equation in standard notation will.
So, if x = 2, it returns 9, and if x = 2.6, it returns 21.9024

The formula, though I don't know if this works for everything, I can get down so far is:

where

or

or some variation or whatever up until x^5, in which case it converges quickly but is not exact.
(Doesn't work when the derivative does not contain x or something. I don't know how to describe it because I'm not in calculus!)


Any help on completing the pattern?
Anyone already know the pattern?

 

[Deleted member 4993]

Firstly, I'm in precalculus, so I don't really know any of the notation
Is there a theorem to find the equation of a summation that would allow me to sum the terms from, say, 1 to 2.6?

For example:

And if x is all real numbers, the equation in summation notation won't return an answer for any non integers, while the equation in standard notation will.
So, if x = 2, it returns 9, and if x = 2.6, it returns 21.9024

The formula, though I don't know if this works for everything, I can get down so far is:

where

or

or some variation or whatever up until x^5, in which case it converges quickly but is not exact.
(Doesn't work when the derivative does not contain x or something. I don't know how to describe it because I'm not in calculus!)


Any help on completing the pattern?
Anyone already know the pattern?

If you are in pre-calc, why are you dealing with problems involving differentials and integrals?

Regarding your question about summing over n as n goes from 1 to x:

The formula that you are using is only valid for "integers"- more specifically consecutive integers beginning with 1 - where n varies as 1,2,3,4.....x (an integer)

 

[srmichael]

If you are in pre-calc, why are you dealing with problems involving differentials and integrals?

Cuz EVERYTHING is done bigger and badder in Texas!!

 

[Gouskin]

The formula that you are using is only valid for "integers"- more specifically consecutive integers beginning with 1 - where n varies as 1,2,3,4.....x (an integer)

But it is valid! The equation that is generated by this formula creates a line who's derivatives have not discontinuities. It would also be incredibly useful. For example, factorials of non integers can be used to calculate sine waves. Factorials are calculated with:

(and various other methods, but forget about those for a second )
But if one could figure out the formula for generating an equation to solve for the summation of this from 1 to a non integer, Sine waves could be calculated in a different way!;

It possibly might even return the taylor series when simplified!

Also, to be very frank, class does not represent level.

 

[JeffM]

Firstly, I'm in prealculus, so I don't really know any of the notation
Is there a theorem to find the equation of a summation that would allow me to sum the terms from, say, 1 to 2.6?

For example:

And if x is all real numbers, the equation in summation notation won't return an answer for any non integers, while the equation in standard notation will.

Because the notation means to add x + 1 - n terms together. So x is constrained. The (x - n) must be a non-negative integer, or else you do not have a definite number of terms to add up. So what do you mean by the equation in standard notation?

So, if x = 2, it returns 9, and if x = 2.6, it returns 21.9024

The formula, though I don't know if this works for everything, I can get down so far is:

where

or

or some variation or whatever up until x^5, in which case it converges quickly but is not exact.
(Doesn't work when the derivative does not contain x or something. I don't know how to describe it because I'm not in calculus!)


Any help on completing the pattern?
Anyone already know the pattern?

.

 

[Gouskin]

Because the notation means to add x + 1 - n terms together. So x is constrained. The (x - n) must be a non-negative integer, or else you do not have a definite number of terms to add up. So what do you mean by the equation in standard notation?.

When I said standard notation, I was thinking that a polynomial was more standard than a sum? Sorry, I haven't "officially" learned any of the notation yet; all I know is what I've taught myself.

But I mean to find what is in between the integer sums.
"else you do not have a definite number of terms to add up" is what I'm trying to get around. Much in the same way that there's a solution to finding non-integer factorials, I'm certain there's a solution to this, and I was wondering if anyone had found an answer.

 

[daon2]

When I said standard notation, I was thinking that a polynomial was more standard than a sum? Sorry, I haven't "officially" learned any of the notation yet; all I know is what I've taught myself.

But I mean to find what is in between the integer sums.
"else you do not have a definite number of terms to add up" is what I'm trying to get around. Much in the same way that there's a solution to finding non-integer factorials, I'm certain there's a solution to this, and I was wondering if anyone had found an answer.

Non-integer factorials are values of the Gamma function, a function that extends the factorial function and which "best" approximates it (see http://en.wikipedia.org/wiki/Bohr–Mollerup_theorem). But how factorials are extended is NOT unique. I can define x! to be |floor(x)|!. That is a valid extention of the factorial function, too. You can even find other ones which are continuous or differentiable.

So you need to decide what your goal is when defining this new kind of sum; exactly what do you wish to accomplish? in what way will your new sum formula be the "best" extension of the standard one?

 

[Gouskin]

Gamma function, a function that extends the factorial function and which "best" approximates it (see http://en.wikipedia.org/wiki/Bohr–Mollerup_theorem).

Oh wow, that equation for calculating non-integer factorials looks like it's a lot better than mine....

Anyway, the purpose is basically to be able to write sums in the form of a polynomial. I've actually expanded the formula to:

(Where the polynomials contain an x raised to a power greater than 0)

Now it's really just a matter of finding an efficient formula to calculate the sequence 12, -720, 30240, -1209600, 47900160, ...
In fact, if you can figure that out, that's all I need.
Thanks.

 

[Gouskin]

Non-integer factorials are values of the Gamma function, a function that extends the factorial function and which "best" approximates it (seehttp://en.wikipedia.org/wiki/Bohr%E2...llerup_theorem)

Oh wait no. That formula is dumb. My formula converges a lot faster and is totally better.
Note that because I'm cool (sarcasm), when I say

I mean

and when I apply a limit, it accounts for the variable it's limiting on all sides of the equation.

Compare this when s = 10 to the Bohr–Mollerup theorem.

Yeah, now that I think about it, somebody is probably going to say, "oh you just took that from Wikipedia," "that's just the Gaussian thing gamma with the A_x thing" so I guess I'll just have to show how I derived it....

Summation is sort of like integration so let's try that:

"Hmm, that looks a lot like a root function. Let's figure out what root it is if any"

"Hmm, it's a square root. sqrt(ax) What's a equal to?"

"It converges on x! as x limits to infinity, so"

 

[daon2]

Oh wait no. That formula is dumb. My formula converges a lot faster and is totally better.

Compare this when s = 10 to the Bohr–Mollerup theorem.

What formula? The definition of the Gamma function? Your expression above also makes no sense. I'm not trying to bring you down, you obviously love mathematics, but calling a landmark theorem about one of the most useful functions "dumb" highlights your maturity.

 

[Gouskin]

What formula? The definition of the Gamma function? Your expression above also makes no sense. I'm not trying to bring you down, you obviously love mathematics, but calling a landmark theorem about one of the most useful functions "dumb" highlights your maturity.

Sorry about insulting the theorem, but I have to ask, how does my expression not make any sense?
...Oh. Sorry, I forgot to put the limit on both sides of the equation. I'm dumb.
If you want to see an interactive version of it, I just made a graph here: https://www.desmos.com/calculator/52q1s6n9l8
The blue line is the factorial function
The green line is my approximation function.
Drag the s slider around to make it more and less accurate.

 

[lookagain]

<------ No, \(\displaystyle \ \dfrac{x^x}{e^x} = \bigg(\dfrac{x}{e}\bigg)^x\)

.

 

[Gouskin]

The exponent is outside of the fraction. I left off the parentheses because I think they're ugly.

 

[lookagain]

The exponent is outside of the fraction. I left off the parentheses because I think they're ugly.

Then what you wrote is wrong. A pair of grouping symbols, such as parentheses, are required.

 

[Gouskin]

Then what you wrote is wrong. A pair of grouping symbols, such as parentheses, are required.

No, it's really not. My teachers do this all the time. What I wrote is correct, what you interpreted it as being is wrong.

 

[lookagain]

No, it's really not. My teachers do this all the time. Then, they are wrong and/or lazy!What I wrote is correct No, it is not., what you interpreted it as being is wrong.I didn't interpret it as wrong. You don't get to type something wrong and then expect someone else to mind-read what you meant.

Either type it correctly, or don't type it wrong/half-assed.

 

[daon2]

I don't think anyone is here to discredit your findings, be them rediscoveries or new approaches on these things. The level of work you are doing is that of advanced undergraduate/graduate level material and it is specialized, so you should probably seek a mentor in analytic number theory who knows this stuff like the back of their hand (free help websites like this will be of little benefit). I'm still not certain if you're a "troll" but if you're not, then you should know (being as smart as you apparently are!) this is beyond first-level classes like calculus, and should talk to your undergraduate adviser immediately.

 

[Gouskin]

Either type it correctly, or don't type it wrong/half-assed.

I can tell you now that your computer will have no problem telling the difference: https://www.desmos.com/calculator/nnhd1fz58k
Anyway, you know, your argument would have come across much more effectively had you not been so blunt at the start. I would have used a concession to persuade me into changing it. You've wasted your time.

But really, if people are going to get this butthurt over such a simple thing, I might as well change it.

 

[lookagain]

I can tell you now that your computer will have no problem telling the difference: https://www.desmos.com/calculator/nnhd1fz58k That site shows the wrong notation. so, you cannot use it as an example of how it could be right.
Anyway, you know, your argument would have come across much more effectively No, there is no "anyway" about it.had you not been > > so blunt < < at the start. I would have used a concession to persuade me into changing it. You've wasted your time. No, you acted stubbornly defiant.

But really, if people are going to get this butthurt over such a simple thing, I might as well change it.

No, you are deflecting onto me. Just change it because it's the correct thing to do. You had better learn more how to argue. And be gracious about accepting the correction.

 

[Gouskin]

I don't think anyone is here to discredit your findings, be them rediscoveries or new approaches on these things. The level of work you are doing is that of advanced undergraduate/graduate level material and it is specialized, so you should probably seek a mentor in analytic number theory who knows this stuff like the back of their hand (free help websites like this will be of little benefit). I'm still not certain if you're a "troll" but if you're not, then you should know (being as smart as you apparently are!) this is beyond first-level classes like calculus, and should talk to your undergraduate adviser immediately.

Thank you for the advice. It appears to be very sound. However, I do not know how to go about seeking out a mentor in analytic number theory. But what's fun about mathematics for me is just questioning things. (Call me crazy but I think that 2/+0 = 2 infinity and I think I can prove it, too. Who should I talk to for that?) Right now I am in the 11th grade and I am therefor not familiar with any undergraduate advisers (Unless I'm just stupid and don't know what an undergraduate / undergraduate adviser is). Any tips on that?
Once again, thank you for your advice. I'll definitely consider it.