Telescoping Sum: All the terms of this series cancel out except for the first and last term.
In this video, I go over an infinite series in which arises an example of the famous telescoping sum such that all the terms of the series cancel except for the first and last term. The series with terms 1/(n(n+1)) is not a geometric series, so we have to start off with the definition of a convergent series and begin by writing out the terms of its n-th partial sum. We can then simplify the terms of the partial sum by using partial fraction decomposition. This yields the terms 1/n - 1/(n+1), which results in all of the middle terms to cancel out in pairs, leaving just 1 - 1/(n+1). Taking the limit as n approaches infinity, we obtain our sum is equal to 1. The telescoping sum gets its name from the collapsing telescope (the ones pirates had).
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Timestamps
- Example 6: Find the sum if this series is convergent – 0:00
- Solution: This is not a geometric series, so go back to the definition of a convergent series and compute the partial sums – 0:17
- Simplify the partial sum via partial fraction decomposition – 1:55
- Notice that all the terms in the partial sum cancel except for the first and last! – 5:05
- This is named after a collapsing telescope (like how pirates had) – 7:49
- The limit of the partial sums is equal to 1 – 8:22
- Graph showing the series terms approach zero as the partial sums approach 1 – 9:44
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