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Note

The ith term of the series (i starts at 0) is 2ⁱ T(n/2ⁱ). The series ends when we hit T(1), or $n/2^i \;=\; 1 \;\longrightarrow\; n \;=\; 2^i \;\longrightarrow\; i \;=\; lg n$ .

T(n) = n + n + n + ... + 2^{lg n} T(1)

= n + n + n + ... + nT(1)

= $\sum_{i=0}^{lg n - 1} n$ + nT(1)

= nlgn + n $\Theta(1)$

= nlgn + $\Theta(n)$

= nlgn + o(nlgn)

= $\Theta(nlgn)$

Do not simplify asymptotic expressions until there is no summation in the expression.

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