How to sum a "complex" finite geometric series?

Question

I'm trying to prove analytically the time complexity of a problem in "Cracking The Coding Interview". The algorithm in question involves calling an O(1) operation on each node of a binary tree once for each level of nodes above it and every node in the tree is visited. The cost of such an algorithm, it seems to me would be:

$2^1(1) + 2^2(2) + 2^3(3) + ... + 2^{log_2 n}log_2(n)$ where n is the number of nodes.

I'm not sure how to evaluate the above using the usual equation for finite geometric series: $a_1\frac{(1-r^n)}{(1-r)}$ where $a_1$ is the first value and r is the constant factor by which the series grows, because the series does not seem to grow at a constant factor; r grows with each step of the algorithm. Does this sort of series have a name?

Sorry, but I’m too lazy tonight to convert your post to readability by applying the relevant MathJax. Your last term, though, does not seem to fit into the pattern of the three terms you wrote out explicitly. — Lubin, Jun 10 '18 at 23:47
Sorry, first post, applied MathJax. The summation would increment the exponent and factor by one up until $log_2(n)$ because that would be the number of levels in a balanced binary tree which is the worst case for this algo I should have mentioned. @Lubin — Allen More, Jun 11 '18 at 00:04
Possible duplicate of Formula for calculating $\sum_{n=0}^{m}nr^n$ — Hans Lundmark, Jun 11 '18 at 07:23

Μάρκος Καραμέρης · Accepted Answer · 2018-06-11T00:14:19.753

2

Hint: Note that $(\sum_{i=0}^{n}x^i)'=\sum_{i=0}^{n-1}(i+1)x^i$ and thus $\sum_{i=1}^{n}ix^{i}=\sum_{i=0}^{n-1}(i+1)x^{i+1}=x(\sum_{i=0}^{n}x^i)'=x(\frac{1-x^n}{1-x})'$. Can you take it from here?

edited Jun 11 '18 at 00:14

answered Jun 11 '18 at 00:09

Μάρκος Καραμέρης

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Got it, thanks. Is there a name for this trick? I feel like I should know this. – Allen More Jun 11 '18 at 00:29
I don’t think there is a standard name but you should find similar methods in any book about combinatronics and generating functions/series. – Μάρκος Καραμέρης Jun 11 '18 at 01:14

score 1 · Answer 2 · answered Jun 11 '18 at 00:37

Consider the geometric formula

$$\frac{r^n-1}{r-1} = 1+r+r^2+r^3+...+r^{n-1}$$

We take the derivative on both sides in terms of $r$:

$$\frac{(nr^{n-1})(r-1)-(r^n-1)}{(r-1)^2}=1+2r+3r^2+...+(n-1)r^{n-2}$$

We then multiply both sides by $r$ in order for the coefficient to match the exponent of $r$:

$$r\frac{(nr^{n-1})(r-1)-(r^n-1)}{(r-1)^2}=r+2r^2+3r^3+...+(n-1)r^{n-1}$$

Substituting $r=2$:

$$2[(n2^{n-1})-(2^n-1)]=2(1)+2^2(2)+2^3(3)+...+2^{n-1}(n-1)$$

At this point you're finished, but you can substitute $n=n+1$ so that $n$ equals the number of terms, and we can also simplify the LHS. The final formula is

$$2^{n+1}(n-1)+2$$where $n$ is the number of terms.

How to sum a "complex" finite geometric series?

2 Answers2