# Sum from infinity to infinity of infinity

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!

• Have you learned about series? Or you just know the summation notation? – Đào Minh Dũng Oct 7 '20 at 2:39
• @ĐàoMinhDũng I have learned about series. – Spencer Lutz Oct 7 '20 at 2:44
• 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. – JMoravitz Oct 7 '20 at 2:50
• 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. – JMoravitz Oct 7 '20 at 2:52
• @JMoravitz Thanks so much! This was helpful. – 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.