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If I have a finite sequence of expressions $a_1+a_2+a_3+....a_k=\infty$, does that imply that at least one such $a_j=\infty$?

I know that if it didn't it would make the sum not equal to infinity, but it still does not make intuitive sense to me. Couldn't part of the magnitude be contained in each expression equally and still be such that one term didn't diverge?

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Have to be careful about meaning of "$\infty$" but roughly speaking yes. –  André Nicolas Dec 18 '12 at 3:42
    
even if the terms are them selves infinite sums –  Ethan Dec 18 '12 at 3:43
    
Yes, it should be OK. But remember that infinite sums do not necessarily behave like finite sums. For example, you cannot rearrange order of summation arbitrarily. –  André Nicolas Dec 18 '12 at 3:48
    
What do you mean by that statement –  Ethan Dec 18 '12 at 3:50
    
I mean, for example, that if you take for example the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$, by rearranging the terms you can make the series to converge to any value you like. –  André Nicolas Dec 18 '12 at 3:56
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Asking whether $a_1+...+a_k=\infty$ implies at least one of the $a_i$ is $\infty$ only makes sense if we have defined addition of $\infty$. If we work with real numbers only then the sum of a finite number of real numbers is certainly a real number.

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Put it this way: the sum of $k$ finite expressions is finite. Prove by induction on $k$.

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Yes; suppose that is not the case. Then we know $a_i\leq\max\{a_1,a_2,\ldots,a_k\}=M<\infty$ by hypothesis. Then $\sum a_i\leq k\cdot M<\infty$, a contradiction. Thus, at least one must be infinity.

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