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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 fn (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}}\\:"

State and prove M, test for uniform convergence of sequence of real valued functions defined on [a, b].

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

Let S = {a1, . . . , ap} be a subset of M, for some p ∈ N∗

. Let (xn)n

be a sequence in S.

(a) Show that there is some j, 1 ≤ j ≤ p and a subsequence (xnk)k

of (xn) such

that xnk = aj for all k ∈ N∗

.

Show that for each a ∈ M, the intersection V of all neighborhoods of a

equals {a} .

Show, by any suitable method, that every finite subset S of M is closed

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

sequence, then 􏰄 anbn may diverge.

Give an example of a convergent series 􏰄 an such that 􏰄 a2n is not convergent

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