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  1. K

    Prove that none of them is prime

    Homework Statement Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n]. Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the...
  2. K

    Fermat's little theorem

    Homework Statement From fermat's little theorem deduce that when p is prime, n^p is equivalent to n (mod p) for all integers n. Homework Equations The Attempt at a Solution I know from Fermat's Little Theorem that , n^(p-1) is equivalent to 1 (mod p), but i don't...
  3. K

    L'hopitals rule?

    Homework Statement Find the limit of f(x)=[xcosx]/[x^3 + 1] as x tends to infinity. Homework Equations The Attempt at a Solution Can i use l'hopitals rule here? Or if not, what are the conditions for f(x) to meet so that i may use l'hopitals rule? Without using l'hop i know...
  4. K

    Converging series

    Homework Statement Show that the following series converges: Homework Equations Sum of (from n=1 to infinity) of [3^n + 4^n] / [3^n + 5^n] The Attempt at a Solution Some help on this question would be much appreciated as i really don't know how to start it. Thanks
  5. K

    Complex Integration

    Homework Statement Evaluate the following intergral: Homework Equations Intergral from gamma of (y)dz, where gamma is the union of the line segments joining 0 to i and then i to i+2 The Attempt at a Solution I have no idea how to do this!
  6. K

    Open sets

    Homework Statement For the following set, state and justify whether or not it is open or closed. Homework Equations A={z| Im z >1} The Attempt at a Solution I know the definition of an open set, but im not sure how to use it to solve this question.
  7. K

    Derivative of function

    Homework Statement Compute the derivative of the following function. Homework Equations f:[1,-1] arrow [-pie/2, pie/2] given by f(x)=sin^-1 (x) The Attempt at a Solution I know that f ' (x)=1/[sqrt(1-x^2)] Im not sure how to include the intervals of pie given, not sure what...
  8. K

    Discussing continuity of a function

    Homework Statement Discuss the continuity of the function f defined for all x belongs to [0,1] by f(x)=x if x is rational and f(x)=x^2 is x is irrational. Homework Equations The Attempt at a Solution I have no idea how to begin this question.....some help would be great thanks!
  9. K

    Limit, as x tends to 0

    Homework Statement Does f(x) tend to a limit as x tends to 0? Homework Equations f(x)=[sinx]/[x^2] The Attempt at a Solution Well i sinx would tend to zero and so would x^2, so would the limit just be zero?
  10. K

    Converging series

    Homework Statement Show that the following series converges: Homework Equations Sum of n=1 to n=infinity of [(n+1)/(n^2 +1)]^2 The Attempt at a Solution I thought about using the ratio test, so if the limit as n tends to infinity of a(n+1) / a(n) < 1, then a(n) - the series - would...
  11. K

    Sequence infinity proof

    Homework Statement Prove that the following sequence (a(n)) has the property that a(n) tends to infinity as n tends to infinity. Homework Equations a(n)=[n+7]/[2+sin(n)] The Attempt at a Solution i tried l'hopitals rule, so i got 1/cos(n).....which wouldnt work. so im not sure...
  12. K

    Runge-kutta formula

    Homework Statement Use the fourth order Runge-Kutta formula to advance the differential equation: dy/dt = y with y(0)=1 forward one step h. That is find y(h). Homework Equations The Attempt at a Solution The Runge-Kutta formula is: x(i+1)=x(i)+h/6 [k(1)+2k(2)+2k(3)+k(4)]...
  13. K

    Supremums and infimums

    Homework Statement Show that for all x,y in R (real numbers), sup{x,y}=1/2(x+Y+|X-Y|), and inf{x,y}=1/2(X+Y-|x-y|) Homework Equations The Attempt at a Solution i know that the supremum is the lowest upper bound and that the infimum is the largest lower bound. However i really...
  14. K

    Proof by induction

    Homework Statement Define the numbers a(0),a(1),a(2),...by a(0)=1, a(1)=3, a(n)=4[a(n-1) - a(n-2)]. for n>=2 Show by induction that for all n>=1, a(n)=2^(n-1) [n+2] Homework Equations The Attempt at a Solution proof by induction is not my stong point, and i really don't know where to...
  15. K

    Proof by induction

    Homework Statement Show that (n+1)^4 < 4n^4 whenever n >= 3 Homework Equations The Attempt at a Solution I need to prove this by induction, so i assume it is true and then prove that when n=n+1 it is also true. so it would become (n+2)^4 < 4(n+1)^4 Im not sure how to...
  16. K

    Supremums 'alpha' and 'beta' problem

    Homework Statement Let A, B,be two non empty sets of real numbers with supremums 'alpha' and 'beta' respectively, and let the sets A+B and AB be defined by: A+B={a+b / a belongs to A, b belongs to B}, AB= {ab / a belongs to A, b belongs to B}. Show that alpha+beta is a supremum of A+B...
  17. K

    Increasing functions

    Homework Statement Let f : R(real numbers) (arrow) (0,infinity) have the property that f ' (x) = f (x) for all x. Show that f is an increasing functions for all x. Homework Equations The Attempt at a Solution I know that if f ' (x) > 0 , where all of x belongs to a,b (not...
  18. K

    L'hopital's Rule for solving limit problem

    The question is: Evaluate the following limit: lim(as x tends to 1) of [(x-1)^3]/(logx) I tried using L'hopital's Rule, so i differentiated the top and bottom of the eqn and i got 3x(x-1)^2, then as x tends to 1 this would tend to 0. I don't think this is correct though, and was wondering...
  19. K

    What does converse mean?

    I had to deduce that sqrt 2 + sqrt 3 was irrational. The question then asks me is the converse true. Im not sure what this means though.
  20. K

    Proof - irrational numbers

    Homework Statement Prove that if x^2 is irrational then x must be irrational. Homework Equations The Attempt at a Solution Maybe do proof by contradiction. I'm not really sure where to start.
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