- #1

Mr Davis 97

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## Homework Statement

Let ##\sum_{n=1}^{\infty}a_n## be a series with nonnegative terms which diverges, and let ##(s_n)## be the sequence of partial sums. Prove that ##\lim_{n\to\infty} s_n = \infty##.

## Homework Equations

## The Attempt at a Solution

This isn't a difficult problem, but I want to make sure my details are right.

Note that ##(s_n)## is an increasing sequence that is not convergent. By the monotone convergence theorem, we can conclude that ##(s_n)## is unbounded above. Since we have a sequence that is increasing and unbounded above and does not converge, we see that ##\lim_{n\to\infty}s_n=\infty## ☐

Here is my problem: is this enough to show that ##\lim_{n\to\infty}s_n=\infty##? In my textbook, the definition of an infinite limit is this: ##\forall M\in\mathbb{R}\exists N\in \mathbb{N}\forall n\ge N, s_n>M##. So is what I said, that the sequence is increasing, unbounded above and does not converge, sufficient to show that this definition is true?