Does multiplying all a number’s roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can’t see if the answer has been addressed either.

Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root, multiplied by the fourth root, multiplied by the fifth root, and on until infinity.

The reason it is a puzzle is that while multiplying a positive number by a number greater than 1 will always lead to a bigger number, the roots themselves are always getting smaller. My intuitive guess is that the product will go towards infinity, it could be that the product is finite.

I also apologize that as a simple recreational mathematics enthusiast, I perhaps did not phrase this question perfectly.

Answer

This is an interesting question, but thanks to the properties of exponentiation, it can be easily solved as follows:
k1kn=k1n1/k=nk11k,
and as the exponent diverges (Harmonic series) the whole expression diverges, as long as n>1.

EDIT: the first equality is definition. For the second equality we need to argue why one can pass to the limit.

In other words: for any fixed N the equality
Nk=1n1/k=nNk=11k,
is valid, but why does it hold that
lim
In this particular case it is a consequence of the fact that the map x\mapsto n^x diverges to +\infty as x\rightarrow +\infty and by abuse of notation we write n^{+\infty}=+\infty

If conversely the exponent would converge to a finite limit, as for example in the case mentioned in the comment of IanF1, then this would be a simple consequence of the fact that the map x\mapsto n^x is continuous.

Attribution
Source : Link , Question Author : glowing-fish , Answer Author : b00n heT

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