Determining Concavity, Derivations, and Proofs
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ONLY Q1,2,4
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Suppose that f(x) has a continuous first derivative for all x in R. Prove that f(x) is concave if and only if....
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Solution Summary
This contains three parts: determining whether given series converge or diverge, deriving a condition for an equation to have a rational root and using that condition in a proof, and a set of three proofs regarding concavity.
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Problem #1
(1) I claim that the series diverges
We can evaluate the complexity of , it is equivalent to . Then we have . So diverges
(2) I claim that the series diverges.
The reason is simple. Its general term and does not approach 0.
(3) I claim that the series converges if and diverges if .
Actually, if , then , so the series convergs.
If , the general term , so the series ...
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