what is telescoping sum?

Question

cggurumanjunath · Answer

http://en.wikipedia.org/wiki/Telescoping_series

kainui · Answer

A telescoping sum is a bunch of terms added together that will add on while subtracting off the previous one... An example might show this better than I can explain it, so that's what I'll show.

$$\sum_{a}^{b}[(x+1)^2-x^2]$$

So here when you start adding it, you'll get:

$$[(a+1)^2-a^2]+[(a+2)^2-(a+1)^2]+[(a+3)^2-(a+2)^2]+...$$

Now you can see when you add up all the terms you'll cancel out the (a+1)^2 in the first term with the -(a+1)^2 in the second term. Similar thing happens for the (a+2)^2 term, etc... So the only terms you'll have left are:

$$\sum_{a}^{b}[(x+1)^2-x^2]=(b+1)^2-a^2$$

Interestingly, yet not very surprisingly, this is a lot like an integral... =)

kainui · Answer

I can go more in depth or provide another example that's more concrete if you like, since this one is a little abstract with the variables. I suggest you try it out though on your own first and you'll see what I'm talking about hopefully.

nincompoop · Answer

how would this then be included into integration?

nincompoop · Answer

thanks by the way

kainui · Answer

Well, an integral is just like a telescoping series in that you are only looking at the end points. Integrals are really just infinite sums, so pretty much everything you learn about with series has some kind of analogue in integrals.

kainui · Answer

$$\int\limits_{a}^{b}xdx$$ is almost the same thing as $$\sum_{a}^{b}x$$

except that of course if we didn't multiply each element of our sum in the integral by dx, it would always be a divergent sum since it's adding up an infinite number of things. Fortunately if you add up an infinite number of infinitely small things, you get something we're all "comfortable" with.