Math Answers

Check the convergence of the sequence defined by 𝑢𝑛+1 = (1 + 1/ 𝑢𝑛 ) , 𝑢1 > 0. Note that this is the sequence associated with the continued fraction expansion of the Golden ratio. 

Check the convergence of the sequence defined by 𝑢𝑛+1 = (1 + 1 𝑢𝑛 ) , 𝑢1 > 0. Note that this is the sequence associated with the continued fraction expansion of the Golden ratio. 

Check the convergence of the sequence defined by 𝑢𝑛+1 = 𝑎/ 1+𝑢𝑛 where 𝑎 > 0, 𝑢1 > 0.

Determine the cardinality of the power set of the set {2,3,x,y,z}

A pendulum of length 𝑙 at an angle 2𝛼. Find the time period T of the pendulum. Also let 𝛼 → 0 and obtain the well-known formula 𝑇 = 2𝜋√ 𝑙 /𝑔 .

Suppose that 𝑓 is differentiable and 𝑓 ′′(𝑎) exists. Prove that 𝑓 ′′(𝑎) = lim ℎ→𝑎 (𝑓(𝑎+ℎ)−2𝑓(𝑎)+𝑓(𝑎−ℎ))/ ℎ^2. Give an example where the above limit exists, but 𝑓 ′′(𝑎) does not exist.

Show that the function defined as 𝑓(𝑥) = { sin 1 𝑥 , 𝑥 ≠ 0 0, 𝑥 = 0 obeys the intermediate value theorem

Let 𝑓: (0,1) → ℝ be a bounded continuous function. Show that 𝑔(𝑥) = 𝑥(1 − 𝑥)𝑓(𝑥) is uniformly continuous. 

State suitable conditions and prove that (𝑓𝑔) ′ = 𝑓𝑔 ′ + 𝑓 ′𝑔.

a)     Determine the Laplace transforms of the following functions

(i) f(t) = t³-2t

(ii) f(t) = sin3t-e^2t

(iii) f(t) = e^2t sinh4t

b) Find the Laplace transform of the following

(i) f(t) = 3e^-4t- 5e^4t

(ii) f(t) = t sin 3t + cos 4t

find the inverse transform of

(i) F (s) = 2/3-1/2s-3

(ii) F (s) = 5s-8/s(s-4), using partial fractions.