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Questions asking us to check whether your solution is correct only are considered off-topic (per http://meta.cs.stackexchange.com/questions/597/). Please pinpoint your doubt and provide *a specific question* to which a meaty answer can resolve your doubt whether your answer is correct or not.

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Not sure if my solution to following recurrence is correct

I have a recurrence relation, it is like the following: $T(e^n) = 2(T(e^{n-1})) + e^n$, where $e$ is the base of the natural logarithm. To solve this and find a $\Theta$ bound, I tried the following …
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