# Asymptotic of $T(n) = T(n-2) + \frac{1}{ \lg n}$ [duplicate]

Trying to determine asymptotic of

$$T(n) = T(n-2) + \displaystyle\frac{1}{ \lg n}$$ $$\lg n = \log_{2}n$$

Last term $\frac{1}{ \lg n}$ give me a lot of trouble. Iterative method doesn't work. Tried change of variables, there is nothing to change.

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## marked as duplicate by Daniel Fischer♦Feb 21 at 21:57

What is the $\lg$ function? Natural logarithm? –  Rod Carvalho Sep 12 '12 at 4:25
Hint: You can bound $T(n)$ between $\frac 12 \sum_{i=4}^n \frac 1{\lg i}$ and $\frac 12 \sum_{i=4}^n \frac 1{\lg (i-2)}$ (to within a small correction), each of which is close to the logarithmic integral $\int \frac 1{\ln x}\; dx$, which goes as $\frac x{ \ln x}$ (like yours to within factors of $\ln 2$)
Might want to check those small values of $i$ in your sums... –  Erick Wong Sep 12 '12 at 4:48