What is the Limit of the Product of Roots as n Approaches Infinity?

In summary, the limit of the product of roots is the value that the product approaches as the roots of a polynomial function approach a certain value. It is significant because it can help us determine the behavior and find the roots of a polynomial function. It can be calculated using the limit definition and algebraic manipulation, and it can have three possible outcomes: a finite value, infinity, or undefined. The limit is closely related to the roots of a polynomial function and can help us determine their number and nature.
  • #1
bobn
22
0
Lt (1 – 1/√2)* (1 – 1/√3)…… (1 – 1/√n+1)
n->∞


There must some simple calculation to find this, but I cannot get this, limit exists since

each tern is less than 1, and limit is more than 0. so limit exists between (1 0).
 
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  • #2
Hi bobn! :smile:
bobn said:
Lt (1 – 1/√2)* (1 – 1/√3)…… (1 – 1/√n+1)
n->∞
…
limit is more than 0 …

Why?

Isn't it less than limn->∞ (1 - 1/2)(1 - 1/3)…(1-1/n) ?
 
  • #3
haa, I am stupid,

1/√n > 1/n implies, (1- 1/√n) < (1 - 1/n).
 
  • #4
thx for reply tiny-tim
 

FAQ: What is the Limit of the Product of Roots as n Approaches Infinity?

What is the limit of the product of roots?

The limit of the product of roots is the value that the product approaches as the roots of a polynomial function approach a certain value. It can be calculated using the limit notation limx→af(x)g(x), where a is the value the roots approach and f(x) and g(x) are the polynomial functions.

What is the significance of the limit of the product of roots?

The limit of the product of roots is significant because it can help us determine the behavior of a polynomial function as its roots approach a certain value. It can also help us find the roots of a polynomial function by setting the limit equal to 0 and solving for a.

How do you calculate the limit of the product of roots?

The limit of the product of roots can be calculated using the limit definition and algebraic manipulation. First, we substitute the value that the roots approach into the polynomial functions. Then, we factor out any common factors and use algebraic techniques, such as rationalizing the numerator or multiplying by the conjugate, to simplify the expression. Finally, we can use the limit laws to evaluate the limit.

What are the possible outcomes of the limit of the product of roots?

The limit of the product of roots can have three possible outcomes: a finite value, infinity, or undefined. A finite value means that the product of the roots approaches a specific number as they approach a certain value. Infinity means that the product of the roots grows without bound as they approach a certain value. Undefined means that the limit does not exist.

How is the limit of the product of roots related to the roots of a polynomial function?

The limit of the product of roots and the roots of a polynomial function are closely related. The value of the limit can help us determine the number and nature of the roots of a polynomial function. For example, if the limit is 0, then the polynomial has at least one root at the value the roots approach. Additionally, the limit can help us find the roots of a polynomial function by setting it equal to 0 and solving for the value the roots approach.

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