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# Prove some theorems on convergence of series

Consider two sequences and whose terms are always positive after some integer (i.e., and for all ). Then define

Prove the following two statements.

1. If there exists a constant with for all , then converges.
2. If the series diverges and if for all then the series diverges.

1. Proof. First, we solve the given relation for (starting with the term so that all terms are positive, hence, nonzero),

Then, since for all , we have

Therefore, the partial sums of are bounded (since is a constant); hence, converges

2. Proof. We are given which implies

This then implies

Since is a constant this implies that if diverges so does

1. Anonymous says:

Just noticed a couple of typos. See suggestions below.

In the proof for part a:
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Then, since for all , we have
\end{frame}

In the proof for part b:
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\end{frame}

2. Anonymous says:

In the proof for part a, should the sentence “Then, since 0 < c_N \leq r for all n \geq N, we have" be "Then, since 0 < c_N \leq r for all n \geq N, we have" instead?

In the proof for part b,