Consider two sequences and whose terms are always positive after some integer (i.e., and for all ). Then define
Prove the following two statements.
- If there exists a constant with for all , then converges.
- If the series diverges and if for all then the series diverges.
- 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
- Proof. We are given which implies
This then implies
Since is a constant this implies that if diverges so does
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
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In the proof for part b:
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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,