A telescoping series is a series in which consecutive terms cancel part of one another, producing a significant simplification when the terms are summed.
The cancelling parts are neighbouring terms in the summation pattern, but they need not be immediate neighbours in the sequence; gaps or intervening terms may remain.
Immediate neighbours
\[\sum_{r=1}^{n}(u_r-u_{r+1})\]
\[\sum_{r=1}^{n}(u_r-u_{r+1})=u_1-u_{n+1}\]
Not immediate neighbours
\[\sum_{r=1}^{n}(u_{r+2}-u_r)\]
\[\sum_{r=1}^{n}(u_{r+2}-u_r)=u_{n+2}+u_{n+1}-u_2-u_1\]
Common Telescoping Series — Partial Fractions
A common telescoping-series form is exposed by first expressing a rational term in partial fractions.
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