Prove Limit of Sequence 1 and Show Binomial Theorem

In summary, in the first conversation, it is stated that if a convergent sequence has a limit of an^2=0, then the limit of an will also equal 0. However, if the limit of an^2=L>0, it is possible for the limit of an to not equal √L. In the second conversation, the binomial theorem is used to show that for any x≥0, (1+x)^n will always be greater than or equal to 〔n(n-1)x^2〕/2 for any positive integer n. To find the counter example in (b), the sequence must satisfy the condition that if the limit of a_n equals √L, then the
  • #1
chiakimaron
9
0
1. (a) If｛an｝ is a covergent sequence and limit of an ^2=0, prove that limit of an=0.



(b) If limit of an ^2=L>0 , give an example to show that limit of an =√L may not be true.



2. For any x≧0, show by binomial theorem, that

(1+x)^n ≧〔n(n-1)x^2〕/2 for any positive integer n.
 
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  • #2
What is your atemp? To help you:

What can you say about the function f(x) = x^2.

If I in (b) tell you that you could actually say that if Lim a_n = sqrt(L) then you actually get Lim a_n^2 = L, what condition does the sequence in (b) have to satisfy to give the counter example.
 

1. What is the definition of a limit of a sequence?

The limit of a sequence is the value that the terms of the sequence approach as the index increases without bound.

2. How do you prove the limit of a sequence?

To prove the limit of a sequence, you must show that for every positive number epsilon, there exists a corresponding index N such that all terms of the sequence after the index N are within epsilon of the limit value.

3. What is the binomial theorem?

The binomial theorem is a mathematical formula that provides a way to expand expressions of the form (a + b)^n, where n is a positive integer.

4. How do you show the binomial theorem?

To show the binomial theorem, you can use mathematical induction or Pascal's triangle to demonstrate the pattern in the coefficients of the expanded expression.

5. How is the binomial theorem used in mathematics?

The binomial theorem is used in various areas of mathematics, including algebra, calculus, and probability. It is particularly useful in simplifying and solving complex equations involving binomial expressions.

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