# Explanation of summation equation

Could somebody please explain the following equation to me?

I have no clue what H represents, nor how theta(ln(n)) - theta(ln(k)) results in theta(ln (n/k))

Any explanation would be appreciated.

Thanks!

• H indicates the harmonic numbers – G Cab Dec 7 '18 at 0:56

The $$H$$ notation used is more a shorthand for the harmonic series' partial sums. More explicitly,

$$H(n) = \text{(the n-th partial sum of the harmonic series)} = \sum_{k=1}^n \frac{1}{k}$$

Edit: You could also say that $$H(n)$$ is the $$n^{th}$$ harmonic number, as the definition is the same.

Going from there, I'm just guessing on what $$\Theta$$ represents from what I could Google up. My guess is that it is a function that means "on the order of" or something of the sort. For that, it might be important to note that we can represent the harmonic series in a sort of "closed" form:

$$\sum_{k=1}^n \frac{1}{k} = \ln(n) + \gamma + \epsilon_n \approx \ln(n)$$

In this context, $$\gamma$$ is the Euler-Mascheroni constant, and $$\epsilon_n$$ is a sort of "error constant" that establishes the equality. It is proportional to $$1/2n$$ and thus $$\epsilon_n \to 0$$ as $$n \to \infty$$.

And then I guess because $$\ln(n) - \ln(k) = \ln(n/k)$$ per a property of logarithms, that might explain your other question.

But I want to emphasize that I'm kinda just guessing here since I'm not particularly familiar with the big-theta notation or how to utilize it. I think that's what's at play here, but it's just a guess.

• Yes, $\Theta$ basically means "on the order of", i.e., bounded both above and below up to a constant. See Big-O-notation – Jair Taylor Dec 7 '18 at 1:05
• This was an absolutely amazing explanation. Thank you so much for taking the time to explain this to me. – Jerry M. Dec 7 '18 at 1:37