Find which of the infinite series converge and which

Question

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 V_n¹ by the quantity of vinegar in the first vessel after n rounds of this game. Likewise, let V_n² be the quantity of vinegar in the second vessel. Show that

V¹_n+1 = V_n¹ - kV_n¹ + kV_n² n+1

V²_n+1 = V_n² - kV_n² + kV_n¹

(b) Use the results of part (a) to show that

V_n+1² - V¹_n+1 = (1 - 2k)ⁿ⁺¹

(c) Finally show that

V_n+1¹ = 1/2(1 - (1-2k)ⁿ⁺¹)

V_n+1² = 1/2(1 + (1-2k)ⁿ⁺¹)