What's wrong with these equations? [duplicate]
My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ $$2x=x$$ $$2=1$$ Yeah! I've succesfully copypasted latex formulas!
I think the problem is in non-formal symbols. It brings me to question, what is the result for $(\sum_{i=1}^{x}x)' = ?$ That's usually an obstacle for those who memorised many things without clear understanding of definitions. So, I'm interested in fundamental mistake of this equations, because I want to get out of this mess)
Solution 1:
the mistake is at the beginning $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ fails miserably when $x$ is not a natural number as the expression $$\overbrace{x+\cdots+x} ^{x\text{ times}}$$ is meaningless for example in case $x=1.37$ what does $$\overbrace{1.37+\cdots+1.37} ^{1.37\text{ times}}$$ mean?
Solution 2:
The first misconception is the definition of $x$-times applied operations.
If one ignores that, there still remains the error in the derivation
$$\frac d{dx}\sum_{i=1}^x x \neq \sum_{i=0}^x \frac d{dx} x = x$$
You have to use Leibniz integral rule, seeing the sum as a special type of integral, namely
$$\frac d{dx} \sum_{i=1}^x x = \frac d{dx} \int_0^x x d\#(i) = x|_{i=x} \cdot 1 - x|_{i=0} \cdot 0 + \int_0^x 1 d\#(i) = x + x = 2x$$
Where $\#$ denotes the counting measure. If you understand this, I can elaborate on the RHS.