Reference no: EM13995717
Short answer question
Use suitable tests to find which of the following infinite series converge and which diverge. Be sure to state which test you are applying.
(a) n=1Σ∞ n/(n3+1) (b) n=1Σ∞ (-1)n(1.01)n
(c) n=1Σ∞ encos2n/Πn (d) n=1Σ∞ 2n/n2
Detailed answer question
You have two 1 litre vessels. The first contains oil while the second contains vinegar. You have two spoons, each can scoop out k litres of liquid, and you use the spoons to transfer k litres between each vessel. That is, you take a spoon from one vessel, a spoon from the other and then you transfer their contents over to the other vessel. You mix the vessels thoroughly so that each vessel contains a mixture of oil and vinegar. This caper is then repeated any number of times.
(a) Let Vn1 by the quantity of vinegar in the first vessel after n rounds of this game. Likewise, let Vn2 be the quantity of vinegar in the second vessel. Show that
V1n+1 = Vn1 - kVn1 + kVn2 n+1
V2n+1 = Vn2 - kVn2 + kVn1
(b) Use the results of part (a) to show that
Vn+12 - V1n+1 = (1 - 2k)n+1
(c) Finally show that
Vn+11 = 1/2(1 - (1-2k)n+1)
Vn+12 = 1/2(1 + (1-2k)n+1)
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