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...
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...
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...
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
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!
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.
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...
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!
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?
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...
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...
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)]...
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...
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...
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...
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...
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...
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...
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.