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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 7: Series basics challenge

# Telescoping series

Telescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze. In this video, we use partial fraction decomposition to find sum of telescoping series. Created by Sal Khan.

## Want to join the conversation?

• I remember that Sal's older video on partial fraction decomposition has a simpler way for solving for A and B:

After multiplying -2/[(n+1)(n+2)] and the fractions by (n+1)(n+2), we have
-2=A(n+2)+B(n+1)
and we solve for A by plugging in n=-1, and solve for B by plugging in n=-2.

Why does Sal solve using a system of equations? •  I suspect that in this case he observed that the B term cancels if you add the equations, so doing it this way is pretty easy too. Plus, seeing a number of different ways to solve problems can be helpful.
• I think the most important question here is how do we know if a given series will be telescopic or not? Do we have to resort to expanding the series for the first few terms? And, if we do that, how do we know that we aren't fooling ourselves into a self sense of security if it the first few terms work out but after perhaps the 100th term things go wrong? • When in doubt you would use an inductive proof. That is, demonstrate that it is correct for the first term (or enough terms to establish the pattern), and then demonstrate that if it's true for term n it has to be true for term n + 1.
• I forgot something. At the start of the video in the top left, what is that 'E' or backwards 3 looking symbol? Name and meaning/purpose? • It's a capital sigma, the greek letter Σ. It stands for summation. The "n = 2" below it tells that the variable n starts with the value 2, and the ∞ above tells that it runs to infinity.

For example:

Σ{n = 1 to 4} n
would mean
1 + 2 + 3 + 4
= 10

Σ{n = -1 to 4} n^2
would mean
(-1)^2 + 0^2 + 1^2 + 2^2 + 3^2 + 4^2
= 1 + 0 + 1 + 4 + 9 + 16
= 31

Σ{n = 0 to ∞} 1/(n + 1)
would mean
1/(0+1) + 1/(1+1) + 1/(2+1) + 1/(3+1) + ...
= 1/1 + 1/2 + 1/3 + 1/4 + ...
= ∞
• Why is it called a telescoping series? • Have you ever seen an old-fashioned telescope made of two or more
tubes that slide inside one another, so you can compactly store it?
It's sort of like this:

o------------o#####o------------o
o------------o ######## o------------o|
o------------o##############o------------o |
| | <----- | |
o------------o##############o------------o |
o------------o ######## o------------o|
o------------o#####o------------o

A telescoping series similarly lets each term "slide" into the next,
"collapsing" the whole thing down to a very simple sum.

Perhaps that configuration is what prompts the name. It does look a
lot like a telescope or tripod leg being collapsed.
• Wouldn't it just be easier to expand the denominator and then take the limit as n goes to infinity? • Allie this was actually a really good question, and after reading it I was wondering the same thing. Eventually i came to the conclusion the reason it doesn't work is because, whilst this may work without the Summation (sigma) when taking a single term to infinite, it doesn't work when we're taking an infinite number of terms (even if they're very small) to infinite, because all of these terms are adding up and possibly converging to some value, in this case 2/3! That's how I understand it anyway.
• Is there actually a formulaic way to recognize this solution/solve for it or do you just have to guess and look for terms to cancel out? • The first part, partial fraction decomposition is a typical way to handle this type of integrand. Actually usually you have a quadratic expression that you try to factor to get the integrand in this form in order to apply PFD). As for the telescoping part, consider this video as an introduction to that type of series - so now you are aware of them and can be on the lookout for them - when things seem to not be falling into place, check if the series is telescopic and apply a similar analysis.

More and more, the math of integration and the math that follows requires you to develop your intuition and creativity to come up with a solution. You will discover that many great proofs happened by a seemingly left field approach. To be able to do this demonstrates that you have a grasp on the implications, consequences and connectedness of the theorems and definitions you are learning.
• What is the link to the video on partial fraction expansion? • I do not understand why the (n+1) is used instead of the (n+2)? They are the factors of (n^2+3n+2) • We aren't simply choosing to use one instead of the other. Sal works through a chain of reasoning to show that in this calculation, all the terms cancel out except for the (n + 1) part of the first term and the (n + 2) term of the last one. Then we see that when we take the limit as n goes to infinity, the first one is unaffected but the second one disappears, so we're simply left with the first (n + 1) term.  