ok, so a factor of n!+3 is 3, and a factor of n!+2 is 2 and so on.
So a factor of n!+n is n. This shows that none of them is prime.
How would i go about doing the second part of the question?
Is (n factorial + 2) a factor of it?
Which would then apply to the others, which would show that none of them were prime.
For them each to be prime, their only factors would have to be themselves and 1.
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| I am z >1}
The Attempt at a Solution
I know the definition of an open set, but I am 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?
i know for the comparison test, you assume that a(n)<= k.b(n)
Then if b(n) is convergent, a(n) is also convergent.
However i still don't see how i can get my series to be <= to a constant times another 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...
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 wouldn't work.
so I am not...
thankyou very much for that...i understand all of what you have said.
So i found for my equation that f[t(0),y(0)] = 1, since y'=y and y(0)=1
Im still slightly confused as to how to find y(h)?
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...
Well the definition of supremum is that it's the least upper bound, it is greater than or equal to each element in the set.
I still don't know how so start off showing that alpha+beta is the sup of A+B...
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.