Find which of the infinite series converge and which

Question

Assignment 1:

Short answer question

Use suitable tests to find infinite series converge and diverge. Be sure to state which test you are applying.

_n=1Σ^∞ eⁿcos²n/Πⁿ 

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)

Assignment 2:

Detailed answer question

(a) Use integration by parts to express

In(x) = sinⁿ(x) dx

in terms of I_n-2(x).

(b) Hence show that

_Π/4∫^3Π/4 1/sin⁴(x) dx = 8/3