Real Analysis Answers

Prove that the sequence {a_n /n } is convergent where { a_n } is a bounded sequence.

Prove that

lim n→∞ [ 1/ √(2n-1) + 1/ √(4n-2²) + 1/ √(6n-3²) +.... + 1/n ] = π /2

Prove that the set of integers is countable.

Examine the convergence of the following series:

i) (3×4)/5² + (5×6)/7² + (7×8)/9²....

ii) 1 + 4x + 4²x² + 4³x³ +....(x > 0)

Prove that the function f defined by

f(x) = -2, if is rational

f(x) = 2, if is irrational

is discontinuous,∀ x ∈ R, using the sequential definition of continuity.

3ⁿ>2n²

Let f [: − 3,3 ] → R be defined by f (x)= 5[x] + x³where [x] denotes the greatest integer ≤ x. Show that this function is integrable.

Prove that any n-th root of unity is a primitive d-th for a uniqued/n ?

Define a partition of an interval. Write any three different examples of

partitions of [0,1].

(ii). Let a real valued function 𝑓 be defined and bounded over [𝑎, 𝑏]. Prove that 𝑓 is

Riemann integrable over [𝑎, 𝑏] if for each 𝜖 > 0 there is a partition 𝑃 such that

𝑈(𝑃, 𝑓) − 𝐿(𝑃, 𝑓) < 𝜖

(iii). Show that 𝑓(𝑥) = 𝑥2

is Riemann integrable over [0,2].

Let f [: − 3,3 ] → R be defined by f (x)= 5x + x³ where [x] denotes the greatest integer ≤ x. Show that this function is integrable.