What is $\sum_{n=∞}^{∞}∞$? Indeterminate? Is the answer the same for $\sum_{n=∞}^{∞}k$ for any non-zero $k$?

This is probably a dumb and also useless question but I was just curious. Thanks!

  • $\begingroup$ Have you learned about series? Or you just know the summation notation? $\endgroup$ – Đào Minh Dũng Oct 7 '20 at 2:39
  • $\begingroup$ @ĐàoMinhDũng I have learned about series. $\endgroup$ – Spencer Lutz Oct 7 '20 at 2:44
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    $\begingroup$ Formally, $\infty$ should not appear as either limit or as the summand. When we say $\sum\limits_{n=1}^\infty a_n$ what we really mean is $\lim\limits_{N\to\infty}\sum\limits_{n=1}^N a_n$. In your case you have $\infty$ appearing both as bottom limit, top limit, and summand. You should rewrite this expression with limits so it is unambiguous whether all of these "infities" are "the same" or not. $\endgroup$ – JMoravitz Oct 7 '20 at 2:50
  • $\begingroup$ Consider for instance $\lim\limits_{N\to\infty}\sum\limits_{n=N+1}^{2N} 1$ which is equal to $N$ for each value of $N$ and so the sum diverges to infinity and compare that to $\lim\limits_{N\to\infty}\sum\limits_{n=N}^{N+1} 1$ which is equal to $1$ for each value of $N$. In both cases, you have the upper and lower limits of the summation are both unbounded, and so might have to a lay person both been equally valid as $\sum\limits_{n=\infty}^\infty 1$ despite giving different results. The point is the upper limit in the first grew much faster than in the second. $\endgroup$ – JMoravitz Oct 7 '20 at 2:52
  • $\begingroup$ @JMoravitz Thanks so much! This was helpful. $\endgroup$ – Spencer Lutz Oct 7 '20 at 3:08

Normally we require the summation variable to be an integer. $\infty$ is not an integer. We allow things like $\sum_{i=-\infty}^\infty$ as a shorthand for $\lim_{m \to -\infty} \lim_{n=\infty} \sum_{i=m}^n$ because it is useful. $\sum_{n=\infty}^\infty$ does not have a meaning that I recognize. If you define it clearly, the question will have an answer.


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