Mathematica Create real function from symbolic sum formula

[Swamp Thing]

In this code, I define a function of x as the sum of the first x integers.

In[7]:= fnSum[x_] := Sum[k, {k, 1, x}]

In[8]:= fnSum[x]

Out[8]= 1/2 x (1 + x)

In[9]:= fnSum[3.5]

Out[9]= 6

I would like now to take the symbolic formula underlying fnSum, and use it with real arguments. How can I assign that formula to a new real function, such that fnSum[3.5] will return the correct non-integer value instead of just 6 ?

Of course, this is just an example -- I would like to know how to do this in general.

 

[Dale]

In your function definition use Evaluate on the right hand side

 

[Delta2]

Let me see if i understand you. You want the function, if $x$ is integer to return $$\sum_{n=1}^{x}n$$ but if x is real you want it to return $$(\sum_{n=1}^{[x]}n)+x-[x]$$

where the symbols [x] denote the integer part of x.

Did i get this right?

 

[Swamp Thing]

Delta2: Not quite...
But I tried Dale's suggestion about "evaluate" and it works. I want it to create the expression x/2(x+1) from the summation formula, but then forget that it is about integers -- it should take any argument like, say, 3.5 and return 7.875

In case you're curious, this is related to my other post here:
https://www.physicsforums.com/threa...sus-integral-from-1-to-0.995310/#post-6409241

 

[Delta2]

Ok well if i understand you now, why not define the function as $\frac{x}{2}(x+1)$ in first place instead of the sum definition?

 

[Swamp Thing]

Because I want to do the same with other sums like x^3, x^5 etc and it would be tedious to type each formula.

Using the Evaluate suggestion, I just do:
fnSum3[x_] = Evaluate[Sum[k*k*k, {k, 1, x}]]

which I might even generalize with a variable power.

 

[Delta2]

Ok i see now thanks. Well its good thing that @Dale was very helpful with his suggestion.

 

[Swamp Thing]

It has taken me over a year to understand the power of Mathematica when it comes to this kind of thing, and now I'm able to ask sensible questions anticipating that neat solutions will exist.

 

[Dale]

It has taken me over a year to understand the power of Mathematica when it comes to this kind of thing, and now I'm able to ask sensible questions anticipating that neat solutions will exist.

Yes, it is very powerful. I have used it for everything from learning physics to helping my wife design a good patchwork quilt.

Do you understand why using Evaluate had the result that it did?

 

[Swamp Thing]

Do you understand why using Evaluate had the result that it did?

Not really.

I looked at the Wolfram page on Evaluate, and it says "causes expr to be evaluated even if it appears as the argument of a function whose attributes specify that it should be held unevaluated."

If I type in Attributes[Sum], it includes "HoldFirst" in the list, i.e. the first argument is to be held unevaluated.

But if I define my own function with OR without evaluate, it doesn't list any attributes, e.g. Attributes[MyFunction] is empty.

In any case, it's not clear how deferring evaluation (or not) would affect whether the argument is to be rounded off or not before calculation.

 

[Dale]

Actually, the important part is not Sum, it is := which is SetDelayed. SetDelayed has the attribute HoldAll, so the entire right hand side is unevaluated and gets evaluated fresh each time the left hand side is called.

When Sum is evaluated, if x is a number then it evaluates the Sum numerically, and if x is a symbol it evaluates the Sum symbolically.

So, without Evaluate the Sum is held unevaluated until the left hand side is called and then it is evaluated either way depending on if the argument is numeric or symbolic.

But with Evaluate the Sum is evaluated immediately, while x is still symbolic. This returns the algebraic expression which is then evaluated when the left hand side is called.