Synonymous series?

Algebra Level 3

In school, Timmy was told to calculate the value of the series

\[ a + a^2 + a^3 + a^4 + \cdots , \]

for some known \(a\). But Timmy misheard, he instead calculated the series

\[ a^{-1} + a^{-2} + a^{-3} + a^{-4} + \cdots . \]

Surprisingly, Timmy gets the correct answer!
Assuming that he did his work correctly, what was his answer?

Submit you answer as -1000 if you think that this is an impossible scenario.


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