Wikipedia:Reference desk/Archives/Mathematics/2021 December 13

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December 13

Ex nihilo, 1

Is there a series whose partial sums are always 0, but which converges to 1?--2A02:908:426:D280:E163:1147:6385:A7B8 (talk) 14:10, 13 December 2021 (UTC)

No, because each term is the difference between two consecutive partial sums, which by your first constraint are both 0. So the series is all 0's and converges to 0. Double sharp (talk) 15:17, 13 December 2021 (UTC)

You might be thinking of Grandi's series (although it doesn't meet your criteria). --Amble (talk) 20:21, 13 December 2021 (UTC)

Another way of looking at this is that, by definition, $\textstyle {\sum _{i=1}^{\infty }a_{i}=\lim _{n\to \infty }\sum _{i=1}^{n}a_{i}}.$ (See the article Series (mathematics).) If all partial sums are $0,$ this means that $\textstyle {\sum _{i=1}^{n}a_{i}}=0$ for all $n.$ So then we can simplify the expression for the infinite sum to $\textstyle {\sum _{i=1}^{\infty }a_{i}}=\lim _{n\to \infty }0.$ The right hand side is obviously $0.$  --Lambiam 22:09, 13 December 2021 (UTC)