Please Explain why 1^infinity is indeterminate.?

Question

^_^/thanks/^_^

Answer 1

1 x 1 x 1 ... is 1. So why is indertiminate?

Answer 2

Indeterminate in this context means that the limit of a function cannot be determined by simply evaluting the function at the limit of its inputs. This does not mean that the function itself is undefined at that point: for instance the function 0^0 is indeterminate, because (x→0, y→0)lim x^y may take on any value depending on the functions x and y, but it is not undefined (0^0 is defined to be 1). Conversely, the value 1/|0| is undefined (because there is no number which when multiplied by zero equals 1, and you can't add such a number without creating formal contradictions in your number system), but it is not indeterminate: (x→1, y→0)lim x/|y| is always ∞, regardless of what the functions x and y are (note that the absolute value function here is very important, since otherwise the limit could be ∞, -∞, or nonexistent, and that would be indeterminate). In the case of 1^∞, this value is undefined (since infinity doesn't appear in the real number system), and also indeterminate. Let x=(1+z/y), where z is some arbitrary constant. Then (x→1, y→∞)lim x^y = (y→∞)lim x^y (because as y→∞, x will necessarily approach 1, so that condition is redundant), and (y→∞)lim x^y = (y→∞)lim (1+z/y)^y = e^z. But note that z may be any arbitrary constant, so clearly 1^∞ may have a limit at any positive real number, thus it is indeterminate. Q.E.D.

Answer 3

You are wrong buddy. Why is one to the power of infinity indeterminate???

1 to the power of anything still equals to 1. Because it is 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 .... x 1. The answer will always remain 1 no matter how many times you multiply.

Answer 4

Infinity is a concept introduced to take care of events during which normal rules of mathematics breakdown. For example let try to divide say 4 by zero.IE dividend=4,divisor=zero, what would be the quotient? What ever value you choose for quotient(however big or small)you will end up getting a zero and will never be able to match the dividend 4. In other words our mathematical division has failed. So we say the quotient is infinity. or an INDETERMINATE that which cannot be determined. Here in this example you want to raise 1to the power infinity. Obviously an expression involving an indeterminate will again result (in this case) to be an indeterminate.

Answer 5

What is infinity? Real analysis handles infinities with quantities getting increasingly large.

Formally: A sequence (a_n) converges to infinity (or diverges) if for all natural numbers M there exists an natural number N such that n>N implies that a_n>M.

This is what 'infinity' is, and notions of 'unbounded quantities' need to be avoided for rigor, even though they may be conceptually understandable.

What does 1^infinity mean?

1^inf = lim n->inf 1^n

It should be clear now that 1^inf = 1.

Formally: Let (a_n) denote the sequence defined by a_n=1^n. For all e>0 there exists a natural number N such that n>N implies that |a_n-1|

For those who wrote that lim n->inf (1+1/n)^n = e, this is of course the correct answer to another question, but this is a different limit.

Answer 6

it's indeterminate since infinity stands for an unbounded limit not a number so 1^infinity is indeterminate

Answer 7

one way to look at this is to write 1 = exp{0}.
then 1^∞ = exp{0 • ∞) and 0 • ∞ is indeterminate.

the answers that say 1^∞ = 1 also have a point, since
1^∞ = (limit as n → ∞ of 1^n) = 1. in the same way, one
could say that 0 • ∞ = 0.

saying that 0 • ∞ is indeterminate means that one cannot adjoin ∞ to the real numbers and get a system with well-defined notions of addition and multiplication.

Answer 8

This is why mathematics is beautyful and so related to phylosophical issues... 1^infty is indeterminate but it aproaches 1 in the limit (how does that sound?). The point is, it will take you an infinite amount of time to find out what 1^infty is; however, the trend states that 1^infty goes to 1. Does that make sense?

Answer 9

Cause infinity is a concept not a number. I can see the logic of the first answer and it makes sense, but real answer can never be determined because infinity does not stop. 1^2, you stop at the second multiple. 1 to infinity implies that you never stop therefore you cannot determine the answer.

Answer 10

theoretically.its not possible since anything involving infinity is incalculable.remember that infinity relates to incalculable.
1^infinity is not an indeterminate.
1^anything will always be1
1^1=1,1^2=1,1^3=1................upto 1^infinity

Answer 11

Infinity is a concept not a number. To evaluate 1^infinity, infinity would need to be finite which it isn't by definition. It is the opposite of finite.