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unscientific
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Homework Statement
The quotient test can be used to determine whether a series is converging or not. The full description is in the attachment.
Homework Equations
The Attempt at a Solution
( i ) Why must they both follow the same behaviour? Even if p ≠ 0, it says nothing about the type of series Ʃu and Ʃv are! How can one so hastily conclude that if one converges, the other must too and if one diverges the other must too?
( ii ) If p = 0, won't it imply that the limit of un = 0 as n approaches infinity? So then series Ʃu must definitely converge. Why did they explain it the other way round (if Ʃv converges, then Ʃu must converge) Isn't Ʃu already known to be converging since limit un = 0 ?
( iii ) If p = ∞, won't it imply the limit of vn = 0 as n approaches infinity? (same case as in part (ii) )
Source: Extracted from Riley, Hobson and Bence " Mathematical Methods for Physics and Engineering " - Chapter 4: Series and Limits