How to solve for this recurrence T(n) = T(n − 1) + lg(1 + 1/n), T(1) = 1?

Question

I got stuck in this recurrence:

T(n) = T(n − 1) + lg(1 + 1/n), T(1) = 1?

for a while and it seems the master method cannot be applied on this one.

I may be wrong, but for me it seems like it is linear. – Yeldar Kurmangaliyev Dec 08 '15 at 04:39 — Yeldar Kurmangaliyev, Dec 08 '15 at 04:39

user1952500 · Accepted Answer · 2015-12-08T04:57:43.493

6

We have:

lg(1 + 1/n) = lg((n + 1) / n) = lg(n+1) - lg(n)

Hence:

T(n) - T(n - 1) = lg(n + 1) - lg(n)

T(n-1) - T(n - 2) = lg(n) - lg(n - 1)

...

T(3) - T(2) = lg(3) - lg(2)

T(2) - T(1) = lg(2) - lg(1)

Adding and eliminating, we get:

T(n) - T(1) = lg(n + 1) - lg(1) = lg(n + 1)

or T(n) = 1 + lg(n + 1)

Hence T(n) = O(lg(n))

edited Dec 08 '15 at 04:57

answered Dec 08 '15 at 04:51

user1952500

I agree for a computer (and I am not the one who has downvoted your answer), but then the equation is fundamentally incorrect. If it should be valid on an algorithmic basic on a _Computer_, it should be rewritten as `T(n) = T(n - 1) + lg(1 + 1/n) + k` where `k` is a constant or `O(1)`. In that case, the solution will turn out to be `T(n) = n * k + lg(n + 1)` or `O(n)`. – user1952500 Dec 08 '15 at 05:03

score 1 · Answer 2 · answered Dec 08 '15 at 05:06

Same answer as the other correct answer here, just proved differently.

All the following equations are created from the given recurrence:

Summing all RHS and LHS in the above equations results in:

Since Log(a) + Log(b) = Log(ab),

T(n) = 1 + Log((n+1)/2)
T(n) = Log(10) + Log((n+1)/2) = Log(5n + 5) assuming that base was 10 and using 1 = Log₁₀10

Therefore T(n) = O(Log(5n + 5)) = O(Log(n))

score 1 · Answer 3 · edited Jun 20 '20 at 09:12

1

It is not linear as some people claim. It is O(log(n)). Here is mathematical analysis:

If you will start unrolling recursion you will get:

if you will do this till the end you will have

or in a short form:

Once you approximate the sum with an integral, you will get:

Finally if you will take a limit x -> infinity:

You will see that the first part is

Which gives you a final solution log(x + 1) which is O(log(n))

edited Jun 20 '20 at 09:12

Community

answered Dec 08 '15 at 05:06

Salvador Dali

The summation has a closed form solution for finite `n` outside the limit in this case. – user1952500 Dec 08 '15 at 05:07
@user1952500 :-) this is embarrasing. Anyway there is nothing wrong with this solution apart from it overcomplicates things. It can also show what to do if there is no close form solution – Salvador Dali Dec 08 '15 at 05:19
no worries. I was about to proceed the same route as the limit allows us to quickly compute the solution mentally. Just tried out the other way :) – user1952500 Dec 08 '15 at 05:21

3 Answers3