Are there any rules relating the sum of a function, and the sum of the inverse of that function?

Question

I'm in university and I've hit summation notation for the first time. I haven't studied series before due to a combination of moving universities (they each thought the other had taught this to me), and when I was in high school this wasn't part of the curriculum. So I have a limited amount of information to draw on. I've found the basic rules: $$\sum_{i=1}^n c = nc$$ $$\sum_{i=1}^n ca_i = c\sum_{i=1}^n a_i$$ $$\sum_{i=1}^n (a_i \pm b_i) = \sum_{i=1}^n a_i \pm \sum_{i=1}^n b_i$$ These are fine. I've been using them in some preliminary worksheets with no problems, but I'm coming up against a few problems that deal with the sum of the inverse of something we've already worked out. For instance, Question 3 asked us to prove by induction that: $$ \sum_{i=0}^n r^i = \frac{r^{n+1}-1}{r-1} \quad \text{for all }n\ge0, r\neq 1$$ I managed that alright, but then Question 4 is: "Using Question 3, show that:" $$\sum_{i=1}^n \frac{1}{2^i} < 2$$

I've managed it after a few hours of mucking around. My first thought was to start at Question 3, make $r = 2$, then inverse the whole sum, but that didn't work. And from plugging in a few values I think I can say $\sum\limits_{i=k}^n \frac{1}{f(i)} \text{ isn't necessarily equal to } \frac{1}{\sum\limits_{i=k}^n f(i)}$. I eventually found it worked when $r = \frac{1}{2}$, but my question is: Are there rules about the sum of the inverse of a function? Is there a quicker way of evaluating these?

There are two more questions in a similar vein, the first asking us to prove through induction that $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ which I managed to do, but the second question built on the first and asked us to prove that $$\sum_{i=1}^n \frac{1}{i}>log_{e}n$$

Although I haven't managed to figure that one out yet, that's not really what I'm asking for help with. I'd just like to know if I'm doing these the hard way, and if there's some easy algebraic rules regarding what happens when you alter the function being summed. What happens when you compare the sum of a function and the sum of the inverse, or log of that function?

I did try to Google rules about this, but I couldn't find anything beyond those first 3 rules at the top. I'd be very grateful for any help with this. Thank you :)

Answer 1

And from plugging in a few values I think I can say $\sum_{i=k}^n \frac{1}{f(i)} \text{ isn't necessarily equal to } \frac{1}{\sum_{i=k}^n f(i)}$.

This is correct. $n/\sum{1\over f(i)}$ is something called the "harmonic mean" of $f(i)$, where $n$ is the number of entries. It's not equal to $\sum f(i)$ in general, in fact it is always less than or equal to ${1\over n}\sum f(i)$, which is the "arithmetic mean" commonly known as the average of $f(i)$. Equality is only when $f(i)$ is the same for all values of $i$.

I'd just like to know if I'm doing these the hard way, and if there's some easy algebraic rules regarding what happens when you alter the function being summed.

Unfortunately there aren't, what you've done so far is correct.

What happens when you compare the sum of a function and the sum of the inverse, or log of that function?

It depends.