Now it’s time to look at a genuinely unique infinite series.
The Telescoping Series!
This type of infinite series utilizes the technique of Partial Fractions which is a way for us to express a rational function (algebraic fraction) as a sum of simpler fractions.
In this case, we are going to change our function into the sum of two
“smaller, easier” fractions, where one is positive, and the other is negative.
Having opposite signs is a significant detail for this series because as we will soon discover, we are hoping that terms will cancel in pairs.
In fact, this is how the series gets its name. The partial sums collapse like an old-fashioned collapsable telescope leaving just two or three terms for us to find the sum of the series, as Paul’s Online Notes nicely explains.
Example of a Telescoping Sum
But another way to think about it is that we can’t see the end of an infinite series, but by using our “telescope” we can see that all the terms are canceling, leaving just the first and last terms of the series.
It’s like being able to see the future!
So together we will review the necessary skills for creating our Partial Fractions, and practice generating a few partial sums to determine that terms are indeed canceling.
And then apply a limit approaching infinity to our remaining terms and find what the series converges to.
Now, it is important to note that if we are just trying to determine if series converges or diverges, then applying the Telescoping Series Test will probably not be our first choice. But, if we are asked to find the sum of the series, and it’s not a geometric series then this is a good test to use!
Telescoping Series Video
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