Why can't this series be looked as a geometric series?

Question

Why can't this series $$\sum_{n=1}^{\infty}\frac1{n^n}$$ be looked as a Geometric Series with $r=\frac1{n}$?

I am looking for answers other than "$r$ is supposed to be a fixed real number from the definition of geometric series".

Thank you.

Answer 1

A geometric series with $r=\frac{1}{n}$ and $a=\frac{1}{n}$ would be

$$ \left(\frac{1}{n}\right)+\left(\frac{1}{n}\right)^2+\left(\frac{1}{n}\right)^3+\cdots=\sum_{k=1}^\infty\left(\frac{1}{n}\right)^k$$

and

$$ \sum_{k=1}^\infty\left(\frac{1}{n}\right)^k\ne\sum_{n=1}^\infty\left(\frac{1}{n}\right)^n $$

Answer 2

Sorry to say, but the answer is "$r$ is supposed to be a fixed real number from the definition of geometric series".

Answer 3

The definition of geometric series is that the ratio of $\dfrac{a_{n+1}}{a_n}$ must be a non-zero constant real number.

Answer 4

You can call anything anyhow to create chaos, but calling this raw a geometric series doesn't give you any profit, because all formulae concerning true geometric series wouldn't give you right result.