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  "body": "\n## Summary\n\nA telescoping series is a series whose terms have the form $a_n = b_n - b_{n+1}$ for a sequence $(b_n)$. Its partial sums collapse: $\\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}$. Thus the series converges if and only if $(b_n)$ converges, and then its sum is $b_0 - \\lim b_n$.\n\n## Remarks\n\nFor example, $\\frac{1}{n(n+1)} = \\frac{1}{n} - \\frac{1}{n+1}$, so $\\sum_{n=1}^{\\infty} \\frac{1}{n(n+1)} = 1$.\nThe finite form $\\sum_{k=0}^{n-1} (f(k+1) - f(k)) = f(n) - f(0)$ is a discrete analogue of the [[fundamental theorem of calculus|0NHGY0N0JB]]. Mathlib states it as `Finset.sum_range_sub`.\n\n## Explanations\n\n### L1 human\n\nIn a telescoping sum, each term cancels part of the next term, like the segments of a telescope that slide into each other.\nOnly the first piece and the last piece survive.\n",
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