Find infinite series converge and which diverge - you mix

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). ∑ (n³ - 1)/(n³ + 1)
      n=1

      ∞
(b). ∑ (-1)ⁿ(1.01)ⁿ
      n=1

      ∞
(c). ∑  (2 + eⁿ)/(3 + πⁿ)

      n=1

     ∞

(d) ∑ 2n/n²
     n=1

Detailed answer question

2. 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 = V¹_n - kV¹_n + kV²_n

V²_n+1 = V²_n - kV²_n + kV¹_n

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

V²_n+1 - V¹_n+1 = ( 1- 2k)^{n +1} 

(c) Finally show that

V¹_n+1 = 1/2(1- (1-2k)ⁿ⁺¹)

V²_n+1 = 1/2(1- (1-2k)ⁿ⁺¹)