How was this summation simplified to 1/i?

Question

How is the summation below simplified?

$$\sum_{j=i+1}^{2i} \frac1{i^2} = \frac{i}{i^2} = \frac1{i}$$

Thanks!

One hopes that this was encountered in a more rich context! Taken alone, this notation is correct, but likely to confuse anyone who encounters it. (Which is to say: don't repeat this in your own writing unless you are making a point in text about the summed term being constant and have a compelling reason to use the variable $i$, which is traditionally used as an index, as that constant) — Milo Brandt, Nov 20 '19 at 00:32
Haha. This was a small part of a question for my stochastic probability class! — Raoul Duke, Nov 20 '19 at 00:41

score 4 · Accepted Answer · answered Nov 20 '19 at 00:31

4

The summation variable is $j$, not $i$, so you can pull $\frac{1}{i^2}$ outside of the summation giving

$$\sum_{j=i+1}^{2i} \frac{1}{i^2} = \frac{1}{i^2}\sum_{j=i+1}^{2i}1 = \frac{1}{i^2} i = \frac{1}{i}.$$

answered Nov 20 '19 at 00:31

Cameron Williams

1 Answers1