Secret age

Algebra Level 4

My class wants to know my Maths teacher's age, but she doesn't want to reveal it...

So, she wrote this on the board, thinking we would be too lazy to solve it:

\[\lim _{ k\rightarrow \infty }{ \sum _{ n=0 }^{ k }{ { \left( \frac { 3966 }{ 3965 } \right) }^{ -\left( n+\log _{ \frac { 3966 }{ 3965 } }{ (2) } \right) } } }, \]

and says the value of this is the year she was born.

Assuming that now is the year 2014 and her birthday has passed, what is her age now? (In years) \[.\]\[.\] By the way, you don't need to fetch a calculator for this. Save your effort :D

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