Arithmetic operations on sequences

In summary, to show that {sqrt(a_n)} converges to sqrt(a), you can use the fact that for some M and all n >= M, there is a positive epsilon such that |a_n - a| < epsilon. By factoring the first factor of the absolute value, you can replace the second factor with a clever substitution.
  • #1
dssmith
1
0

Homework Statement



If the sequence {a_n} n=1 to infinity converges to (a) with a_n >0 show {sqrt(a_n)}
converges to sqrt(a)

Homework Equations



hint: conjigate first

The Attempt at a Solution



abs[ (a_n-a) / (sqrt(a_n)+sqrt(a) ) ] < epsilon

i do not own LATEX, yet.
 
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  • #2
You know that for some M, and for all n >= M, there is a positive epsilon such that |an - a| < epsilon.

I would approach this by factoring |an - a| into [itex]|\sqrt{a_n} - \sqrt{a}||\sqrt{a_n} + \sqrt{a}|[/itex]. Then maybe you can replace the second factor by something clever.
 
  • #3
Clever like WHAT?? Waaaaah :'(
 

1. What are arithmetic operations on sequences?

Arithmetic operations on sequences involve performing mathematical calculations on a sequence of numbers, such as adding, subtracting, multiplying, or dividing the numbers in the sequence.

2. What is the purpose of performing arithmetic operations on sequences?

The purpose of performing arithmetic operations on sequences is to analyze and manipulate data in a systematic and organized manner, allowing for easier interpretation and drawing conclusions.

3. What are some common types of arithmetic operations used on sequences?

Common types of arithmetic operations used on sequences include finding the sum, product, difference, and ratio of the numbers in a sequence. Other operations may include finding the average, median, or mode of a sequence.

4. How can arithmetic operations on sequences be useful in real-world applications?

Arithmetic operations on sequences are useful in various real-world applications, such as finance, statistics, and science. They can help analyze trends, make predictions, and solve problems.

5. What are some tips for performing arithmetic operations on sequences accurately?

To perform arithmetic operations on sequences accurately, it is important to carefully follow the order of operations, properly handle negative numbers, and double-check calculations for accuracy. It can also be helpful to use a calculator or computer program to assist with complex operations.

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