# What Are the Limits of {xn} and {yn} in the Given Sequence?

• huey910
In summary, the problem involves finding the limits of two sequences, {xn} and {yn}, defined by recursive equations involving x0 and y0. The answer is lim{xn} = lim{yn} = sqrt(x0y0). To find these limits, it is necessary to show that x_n is decreasing and has a lower bound, while y_n is increasing and has an upper bound. This can be done by induction and using the given equations. It is also mentioned that this is a well-known fact, but no further explanation is given.
huey910
Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
xn+1 =xn +yn , yn+1 = 2xnyn .Find the limits of {xn} and {yn}.

why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?

The problem as stated is wrong (e.g. x0 = 2, y0 =1. they just keep getting bigger).

Oh! Sorry, I transcribed the question erroneously. It should be:

Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
xn+1 =(xn +yn)/2 , yn+1 = 2xnyn/(xn + yn) .Find the limits of {xn} and {yn}.

why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?

1) Show that x_n is decreasing, and y_n is increasing.
2) Show that x_n has a lower bound, and y_n has an upper bound.

Hint: Show that x_n < y_n by induction, and then that x_n+1 < x_n, and y_n+1 > y_n.

Do you know why they must converge in this case? It is a well known fact.

3) Find the limits by using the equations.

It is a well-known fact? Please expand on it and enlighten me! Thank you for the hints though!

## What is the invariance principle?

The invariance principle, also known as the principle of relativity, states that the fundamental laws of physics remain the same in all inertial reference frames.

## Why is the invariance principle important?

The invariance principle is important because it allows us to understand and predict physical phenomena without being limited by a specific reference frame. It also forms the basis for the theory of special relativity.

## Who first proposed the invariance principle?

The invariance principle was first proposed by Galileo Galilei in the 16th century and later expanded upon by Sir Isaac Newton in the 17th century.

## What are some examples of the invariance principle in action?

One example of the invariance principle is the speed of light remaining constant in all inertial reference frames, as demonstrated by the famous Michelson-Morley experiment. Another example is the conservation of energy and momentum in collisions, which remain the same regardless of the reference frame.

## How does the invariance principle relate to the theory of relativity?

The invariance principle is a key concept in the theory of relativity, specifically in special relativity. It forms the basis for the idea that the laws of physics are the same for all observers, regardless of their relative motion. This led to the groundbreaking concepts of time dilation and length contraction.

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