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Telescoping Series theorem vs. Grandi's series
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[QUOTE="Ray Vickson, post: 5664521, member: 330118"] If your book really says that the series converges if and only if ##\lim_{n \to \infty} b_n## exists, then your book is wrong. In fact, the series converges if and only if ##\lim_{n \to \infty} b_n = 0##. That is easy to see: for the partial sum ending with ##(b_{n-1} - b_n)## the value of the sum is ##b_1 - b_n##. For the partial sum with one more term the sum is equal to ##b_1 + \cdots +(b_{n-1} - b_n) + b_n = b_1##. Thus, the partial sums alternate between ##b_1## and ##b_1 -b_n##, so have a limit as ##n \to \infty## if, and only if ##b_n \to 0##. [/QUOTE]
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Telescoping Series theorem vs. Grandi's series
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