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    Convergence of alternating series

    1. Homework Statement Let s be the sum of the alternating series \sum(from n=1 to \infty)(-1)n+1an with n-th partial sum sn. Show that |s - sn| \leqan+1 2. Homework Equations I know about Cauchy sequences, the Ratio test, the Root test 3. The Attempt at a Solution I really have...
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    Finding mean and variance

    1. Homework Statement Suppose that, on average, 70% of graduating students want 2 guest tickets for a graduation ceremony, 20% want 1 guest ticket and the remaining10% don't want any guest tickets. (a) Let X be the number of tickets required by a randomly chosen student. Find the mean and...
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    Proving that f is bounded on R

    1. Homework Statement Suppose that f: R -> R is continuous on R and that lim (x -> \infty+)(f(x) = 0) and lim (x -> \infty-)(f(x)=0). Prove that f is bounded on R 2. Homework Equations I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to...
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    Proving that g1,g2,g3 are linearly independent

    1. Homework Statement Let V = {differentiable f:R -> R}, a vector space over R. Take g1,g2,g3 in V where g1(x) = e^{}x, g2(x) = e^{}2x and g3(x) = e^{}3x. Show that g1, g2 and g3 are distinct. 2. Homework Equations If g1-g3 are linearly independent, it means that for any constant, k in F...
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    Proving lim (as n -> infinity) 2^n/n! = 0

    1. Homework Statement Prove that lim n \rightarrow\infty 2^{}n/n! = 0 2. Homework Equations This implies that 2^{}n/n! is a null sequence and so therefore this must hold: (\forall E >0)(\existsN E N^{}+)(\foralln E N^{}+)[(n > N) \Rightarrow (|a_{}n| < E) 3. The Attempt at a...
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    Correlation and independence in Probability

    1. Homework Statement Let X be a random vairable which can only take three values: -1,0,1 and they each have the same probability. Let Y also be a random vairable defined by Y = X2. Show that i) X and Y are not independent ii) X and Y are uncorrelated 2. Homework Equations To show...
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    First order differentials

    1. Homework Statement Solve: (2t+x) dx/dt + t = 0 2. Homework Equations y' +p(X)y = q(x) and y(x) = (\intu(x)q(x) + c)/u(x) where u(x) = e\intp(x)dx Note this u(x) is 2 to the power of the integral of p(x) 3. The Attempt at a Solution (2t+x) dx/dt + t = 0 becomes: dx/dt + t/(2t+x) = 0 by...
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    Proof by induction

    1. Homework Statement Prove that (\foralln in the set of Natural numbers )[(n \geq 9) \Rightarrow (2n > 4n2 + 1)] 2. Homework Equations To do proof by induction you must first prove for n = 1, then assume true for n and then show for n+1 3. The Attempt at a Solution So for n=1 i have...
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    Marginal/Conditional Probability Mass Functions

    1. Homework Statement A bag contains four dice labelled 1,...,4. The die labelled j has j white faces and (6-j) black faces, j = 1,...,4. A die is chosen at random from the bag and rolled. We define X = the number labelling the chosen die. Y = {0 if the face showing on the die is black; 1 if...
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    Finding x = ymod(n) for x=2^71 and n=23

    1. Homework Statement In this statement find the integer y such that x == y mod(n) where 0 =< y < n x=2^71 n=23 2. Homework Equations Let p be a prime number and let a be an integer which is not divisible by p. Then a^(p-1) == 1mod(p) 3. The Attempt at a Solution I am really struggling...
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    Analysis: proving inverses involving sets

    Let f: X → Y and A is a subset of Y and B is a subset of Y. Prove that: a) f⁻¹(A union B) = f⁻¹(A) union f⁻¹(B) b) f⁻¹(A intersetion B) = f⁻¹(A) intersection f⁻¹(B). I know that f⁻¹(A) = {x ε X : f(x) ε A} and so f⁻¹(B) {x ε X : f(x) ε B} but after that I really don't understand how to prove...
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