Real Analysis Answers

Check, whether the collection G, given by

G' =. { ] 1/(n+2), 1/n [ : n ∈ N}

is an open cover of ]0,1[

Check wether the collection G, given by:

G’ = {]1/(n+2), 1/n[ : n∈N}

is an open cover of ]0,1[.

Calculate the following integrals by using the integration

methods.

1

a) ∫ (4𝑥^34 + 8𝑥^3 + 15𝑥)𝑑𝑥 / (√𝑥^2 + 4x)

0

b) ∫ 𝑑𝑥 / sin 𝑥 ∙ cos^2 𝑥

Suppose 𝑓 and 𝑔 are continuous functions on [𝑎, 𝑏] and that

𝑔(𝑥) ≥ 0 for all 𝑥 ∈ [𝑎, 𝑏]. Prove that there exists 𝑥 in [𝑎, 𝑏] such that

𝑏 𝑏

∫ 𝑓(𝑡)𝑔(𝑡)𝑑𝑡 = 𝑓(𝑥) ∫ 𝑔(𝑡)𝑑𝑡.

𝑎 𝑎

Find the interval of convergence of the power series

∞

Σ [(−1)^𝑛 (𝑛 − 2) / 𝑛^2 . 2^𝑛] (𝑥 − 2)^𝑛

𝑛=3

Prove that the following series is convergent for all 𝑟 ∈ ℝ

Σ (1 + 1/2 + ... + 1/n) (sin (nr) / n).

Show that the series summation of 1/a^2 from a=1 to infinity converge and find the sum of the series. Hence or otherwise prove the summation from a=1 to infinity is equal to π^2/6

Given the function g(x)=(x+7^3

a) Find the critical points of g(x)

b) On what intervals is g(x) increasing and decreasing

c) At what points, if any does g(x) local and absolute minimum and maximum values?

Prove that if g is monotonic on [a,b], then the set of points [a,b] at which g is discontinuous is at most countable.