Real Analysis Answers

Find lim x tends to 0 1-cos²x /x²sinx²

Test the following series for convergence 

∑
n=1
∞ [✓n^4+9 -✓n^4-9]

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

Show that x^4 + x^3 − 4x + 1 = 0 has a root in (0,1)

State Bonnet’s mean value theorem for integrals. Apply it to show that:

|₃∫⁵ cosxdx/x|≤ 2/3

Test the series:

_n=1 ∑∞ (-1)^n-1 [Sin(nx)]/n√n

for absolute and conditional convergence.

Give an example of a series  ∑a_n such that  ∑a_n is not convergent but the sequence (a_n) converges to 0.

Are the following statements true or false? Give reasons for tour answers.

a) −2 isalimitpointoftheinterval ]−3,2].

b) The series (1/2) - (1/6) + (1/10) (−1/4) +.... is divergent.

c) The function, f (x) = sin²x is uniformly continuous in the interval [0,π].

d) Every continuous function is differentiable.

e) The function f defined on R by

f(x)= {0, if x is rational and 2, if x is irrational

Is integral element in the interval [2,3].

Show that the series n=1∑∞ x/(1+n²x²) is uniformly convergent in [α,1] for any α>1.