Real Analysis Answers

Show that the function f defined by f(x) =1//x^2 is uniformly continuous on [5 ,∞].

a) Does the sequence (3+(-1)ⁿ) converge to 2? Justify.

 b) Show that "\\lim _{x\\to \\infty }\\left(\\frac{x-3}{x+1}\\right)^x=e^{-4}"

c) Check whether the sequence f_n (x) = "\\frac{3x}{1+nx^2}" where x ∈ [2,∞ [ is uniformly 

convergent in [2,∞ [

a) Find "\\ lim_{x\\to 0}\\frac{\\left(tanxsec^2x-x\\right)}{x^3}"

 b) Examine whether the equation, x³- 11x +9 =0 has a real root in the interval [0,1]

 c) Check whether the following series are convergent or not (4) 

 i) "\\sum _{n=1}^{\\infty }\\:\\frac{\\left(3n-1\\right)}{7^n}"

(ii) "\\sum _{n=1}^{\\infty }\\frac{\\left(\\:\\sqrt{n^2+3}-\\sqrt{n^2-3}\\right)}{\\sqrt{n}}\\:"

If the distribution is not normally distributed and the sample size is small ,n=10, is the t-test still apropriate to use?explain your answer

) Give an example to show that if the convergence of 􏰄 an is conditional and (bn) is a bounded

∞

sequence, then 􏰄 anbn may diverge.