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# Help if you can, Please.

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I have come across something that I have avoided for a long while, but I must come to terms with it today. My question is how do I calculate a product and/or sum with fractional bounds? An example would be $$\prod_{i=0.5}^{1.5}(x^i)$$ and $$\sum_{i=0.5}^{1.5}(x^i)$$.

Tetration is a hobby of mine. It is so far the most interesting operation I have come across (don't worry this has a connection to what I just described above). If you don't know, tetration is basically the operation above exponentiation. The same way exponentiation is iterated multiplication, tetration is iterated exponentiation. It is known for its INSANE growth. $${}^13=3, {}^23=27, {}^33=7625597484987, {}^43\approx1.258e+3638334640024$$

I was able to define the derivative of tetration as $$\frac{d}{dx}({}^yx)=\frac{1}{x}(\sum_{n=0}^{y-1}(\prod_{k=y-n-1}^y({}^kx)ln(x)^n))$$. And for those who looked into tetration, you will know that non-integer values are not defined well. This is where the question comes into play. Let me say I want to find $$\frac{d}{dx}({}^{1.5}x)$$. What happens to the answer is this: $$\frac{1}{x}(\sum_{n=0}^{1.5-1}(\prod_{k=1.5-n-1}^{1.5}({}^kx)ln(x)^n))$$, which equals $$\frac{1}{x}(\sum_{n=0}^{.5}(\prod_{k=.5-n}^{1.5}({}^kx)ln(x)^n))$$.

There is a sum with a fractional upper bound and a product with both bounds having a fraction. I have no idea how to represent this so if you have any ideas, they could be very helpful. Thanks in advance.

This is useful because if the derivative of tetration is true for non-integer values as well, then the non-integer values of tetration would be able to be defined.

Nov 9, 2018
edited by creepercraft97T3  Nov 9, 2018

### 4+0 Answers

#1
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the sum and product symbols are not defined to use fractional limits.  The index is only ever an integer.

what you can do is multiply the limit by whatever integer is in the denominator of the limit, in this case 2

and then divide the index by that scale factor in the body of the product term

Nov 9, 2018
edited by Rom  Nov 9, 2018
#2
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I don't see a limit in his question, am I missing something?

Guest Nov 9, 2018
#3
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Oh. Thanks anyways.

creepercraft97T3  Nov 9, 2018
#4
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the number at the top of the summation... the last index of the sum

Rom  Nov 9, 2018