Real Analysis Answers

If the power series {summation} a_n xⁿ converges uniformly in ] α ,β [ then so does {summation} a_n (-x)ⁿ . true or false ? Justify

Check whether the series {Summation} n²x⁵/(n⁴+x³) , x belongs to [0, a] is uniformly convergent or not ,where a belongs to R

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.

Prove that -x*-x =x^2

Show that for any A > 1 and any positive k, "Limit" as n approaches infinity "n^k\/A^n=0"

Let ((Xn, Yn)) and ((Un, Vn)) be sequences in R2, and let

(X0, Y0), (U0, Vo) belong to R2.

(i) If (Xn,Yn) converges to (X0, Y0) and (Un, Vn) converges to (Uo, Vo), then (Xn,Yn)+(Un, Vn) converges to (X0,Y0) + (U0,V0) and (Xn,Yn).(Un,Vn) converges to (Xo,Y0)(Uo, V0).

ii) If (Xn, Yn) converges to (X0, Y0), then for any r belonging to R, r(Xn,Yn) converges to r(X0,Y0).