Telescoping series

In mathematics, telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with a succeeding or preceding term.

Related Topics:
Mathematics - Series

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For example, the series

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:sum_{n=1}^infty rac{1}{n(n+1)}

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simplifies as

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:

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sum_{n=1}^infty rac{1}{n(n+1)} = sum_{n=1}^infty rac{1}{n} - rac{1}{(n+1)},

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::::= left(1 - rac{1}{2} ight)

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+ left(rac{1}{2} - rac{1}{3} ight) + cdots,

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::::= 1 + left(- rac{1}{2} + rac{1}{2} ight)

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+ left( - rac{1}{3} + rac{1}{3} ight) + cdots = 1. ,

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While telescoping is a neat technique, there are pitfalls to watch out for:

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:0 = sum_{n=1}^infty 0 = sum_{n=1}^infty (1-1) = 1 + sum_{n=1}^infty (-1 + 1) = 1,

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is not correct because regrouping of terms is invalid unless the individual terms converge to 0. The way to avoid this error is to find the sum of the first N terms first and then take the limit as N approaches infinity:

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:

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sum_{n=1}^N rac{1}{n(n+1)} = sum_{n=1}^N rac{1}{n} - rac{1}{(n+1)},

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::::= left(1 - rac{1}{2} ight)

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+ left(rac{1}{2} - rac{1}{3} ight) + cdots

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+ left(rac{1}{N} - rac{1}{N+1} ight),

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::::= 1 + left(- rac{1}{2} + rac{1}{2} ight)

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+ left( - rac{1}{3} + rac{1}{3} ight) + cdots

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+ left(-rac{1}{N} + rac{1}{N} ight) - rac{1}{N+1} ,

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::::= 1 - rac{1}{N+1} o 1 mathrm{as} N oinfty.,

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