Consider A = 1 + 2 + 3 ... +infinity and B = 1x1 + 2x2 + 3x3 ... + infinity squared. Is B greater than A?

Answer 1

It is a trick question. What is 2 times infinity? The answer is meaningless because infinity is not an amount - it is an idea to express things like the result of one divided by zero. You can say the result is infinity, but that is really meaningless. It is really a sign that your calculations blew up, and your result is meaningless...

Ron.

Answer 2

I suspect my argument is not going to be well received. I say the probability is 50% Let a ∈ℝ, Let b ∈ℝ and randomly select the values for a and b. As already noted, for a ≤ 0, P( a < b²) = 1, this is trivial. Only slightly less trivial is the idea that P(a < 0 ) = 1/2 and thus P( a < b² | a ≤ 0) = 1 and P( a < b² ) ≥ 1/2 Now consider what happens when a > 0 For a > 0, while it is easy to show there is a non zero probability for a finite b, the limit, the probability is zero. a < b² is equivalent to saying 0 < a < b², remember we are only looking at a > 0. If this a finite interval on an infinite line. The probability that a is an element of this interval is zero. P( a < b² | a > 0) = 0 As such we have a total probability P( a < b² ) = P( a < b² | a ≤ 0) * P(a ≤ 0) + P( a < b² | a > 0) * P(a > 0) = 1 * 1/2 + 0 * 1/2 = 1/2 Remember, this is because of the infinite sets. No matter what type of interval you draw on paper or on a computer you will find a finite probability that appears to approach 1. But this is due to the finite random number generators on the computer and if we had this question asked with finite values there would be a a solution greater than 50%. I don't mean to be condescending, but please explain why using the Gaussian to approximate a uniform distribution is a good idea? Aren't infinite numbers fun. Cantor when mad working with them! :)

Answer 3

This is really can't be answered since the question is poorly worded. Some questions that can be answered are:

1. Let A(n) = 1 + 2 + ... + n; and let B(n) = 1^2 + 2^2 + ... + n^2. Find lim (B(n)/A(n)) as n tends to inf.

B(n)/A(n) = [n(n+1)(2n+1)/6]/[n(n+1)/2] = (2n+1)/3 which tends to infinity as n tends to infinity

2. Consider the set A of partial sums: A = {1, 1+2, 1+2+3, ...} and the set B of partial sum of squares: B = {1^2, 1^2+2^2, 1^2+2^2+3^3,...}. Do the two sets have the same cardinality. The answer is "yes" since both sets can be put into a 1-1 relationship with the natural numbers.

Math Rules!

Answer 4

For all values after 1, B will always be greater than A, even to infinity. Infinity signifies that the equation continues on forever for all real positive numbers. You can plug in values higher than 3 to test it.

For example, check out the 50th term in each.
A = 1 + 2 + 3 ... + 50
B = 1x1 + 2x2 + 3x3 ... + 50x50

It's evident that the value of B is in fact greater than the value of A at that time. This trend will continue as you continue toward infinity.

We can conclude that at any instant, B will be greater than A, but since they are both headed toawrd infinity their final value is equal.

Answer 5

B is not greater than A but instead A does equals to B.
As a rule, x + infinity is always equal to infinity.
This is becuase infinity is only a concept to indicate that something goes on and on forever, and never a number.

Therefore, 1 + 2 + 3 + ... + infinity = infinity
and 1^2 + 2^2 + 3^2 + infinity^2 = infinity.

Answer 6

You can say the nth term in a will be (n(n+1))/2. You can also say the nth term in b will be (n^3 /3 ) + (n^2 /2 ) + (n/6). Factor out an n and assume this is true (b>a) and we find (2n^2+3n+1)/3>n-1. More simplification brings 2n^2+3n>3n-4. This brings 2n^2>-4. n^2 is always positive and two is positive and positive two times positive n^2 will always be positive. Any positive number will be greater than -4. meaning no matter what the above statement is true and therefore the entire thing (b>a) is true. Always, unless the rules of algebra break down when using infinity.

Answer 7

A goes to infinity
B goes to infinity ^ 2
infinity * infinity = infinity
therefore, B also goes to infinity
A compared with B
to get whether A or B is greater, just do subtraction
A-B. If it's a negative value, then A is smaller...
so, the answer goes to what is infinity - infinty

Answer 8

There's a problem with the question. Infinity is not a number, so the concepts of "less than", "greater than", and "equal" do not apply, as they apply to numbers.

So technically, the answer is "no," B is not greater than A. Nor is it less that A, nor is it equal to A.

Answer 9

Let us assume that A and B are sum upto the first 'n' terms where 'n' is very large.
A = n(n + 1)/2
B = n(n + 1)(2n + 1)/6

Now, A - B = n(1 - n^2)/3

As, n is very large so n > 1
So A - B is negative.

But, Lim (A - B) as n -> inf = (inf - inf^3)/3 = (inf - inf)/3 = 0

So A = B

Answer 10

yes
as the the square of no is always greater than the no being squared. this rule is also in infinity.