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I know how you can prove that the sequence of terms of a convergent series tend to zero by writing the term as the difference of series to $n$ and $n-1$. However, I would specifically like to know how to prove this via contrapositive/contradiction by first assuming that the sequence of terms doesn't tend to zero and showing that the series doesn't converge. I wrote out that this means we assume that there exists an $\epsilon>0$ such that for all $N \in \Bbb{N}$ there exists $n$ with $n \geq N$ such that $\vert a_n \vert \geq \epsilon$ but I don't know how to get to the conclusion this way. Could someone help write out the proof?

I don't know how to write mathematical notation so sorry about that.

homosapien
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Max123456789
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  • You have proved that an infinite number of terms exist which are greater than $e$ in absolute value. One potential conclusion from this is that the series cannot converge absolutely. A separate conclusion might be that for any given $L$ the series cannot converge to it since for any partial sum where it is close there is another term which must move the sum away again. – abiessu Jan 16 '24 at 21:54

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We can use the fact that the real numbers are complete - so every convergent sequence is Cauchy and vice versa.

Then if we define the partial sums $s_n = \sum_{i = 1}^n a_i$, we have $|s_{n+1} - s_n| = |a_n| > \epsilon$, hence the sequence of partial sums is not Cauchy and thus is not convergent.

If you don't want to use the Cauchy property then you can extend the proof by including a second layer of contradiction (which I think essentially does the same thing as just doing the direct proof, but it's still valid). Assume that the sequence $s_n$ is convergent. Then it has some limit $L$ such that $|s_n - L| < \epsilon'$ (being careful since it's a different $\epsilon$ to the one applied to the $a_n$ terms). Then through careful choice of $N$ and $\epsilon'$ you should be able to show that $|s_{n+1} - s_n|$ must be simultaneously small and big in some fashion. (As a hint, $|s_{n+1} - s_n| = |(s_{n+1} - L) - (s_n - L)|$, and you can apply the triangle inequality.)

ConMan
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