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What is the asymptotic notation of

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i.e.

sum_(i = 1)^n (1 / i)

First of all, this is not a homework. Second, since there is no formula to calculate fraction, I don't know how to express this summation with n and get the asymptotic notation.

That might be Theta(n), but I am not sure.

Zabuzard
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flower
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1 Answers1

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Your expression

sum_(i = 1)^n (1 / i)

is the n-th harmonic number.

From the Wikipedia article:

The n-th harmonic number is about as large as the natural logarithm of n. The reason is that the sum is approximated by the integral:

int_1^n (1 / x) dx

whose value is ln(n).

They also give this nice image which shows the correlation:

correlation between harmonic and natural log

So it is Theta(log n).

See Finding Big O of the Harmonic Series or Simple proof of showing the Harmonic number Hn=Θ(logn) for details of the proof, it is quite simple.

Zabuzard
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