Use the definition of convergence to prove that the lim (1/2)^n=0

In summary, the definition of convergence states that for a given sequence, a_n, and a limit L, there exists a number N such that for any ε>0 and any n>N, the absolute value of (a_n - L) is less than ε. To prove that lim n→∞ (1/2)^n=0, we can use this definition and choose N to be the next largest integer greater than ln(ε)/ln(1/2). This guarantees that for any n>N, (1/2)^n will be smaller than ε.
  • #1
janewaybos
3
0

Homework Statement



Use the definition of convergence to prove that lim n→∞ (1/2)^n=0

The definition of convergence says |a_n-L|<ε


Homework Equations





The Attempt at a Solution



As I understand it:

|(1/2)^n-0|<ε

|(1/2)|^n<ε

then I need to solve for n?

n>(ln(ε))/ln(|1/2|)

Then I choose N=(ln(ε))/ln(|1/2|) but I don't understand why.

n>N>(ln(ε))/ln(|1/2|)??

Given that ε>0 and n>N

then n>(ln(ε))/ln(|1/2|)

then solving for ε I get the statement

|(1/2)|^n<ε

from above. Thanks!
 
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  • #2
janewaybos said:

Homework Statement



Use the definition of convergence to prove that lim n→∞ (1/2)^n=0

The definition of convergence says |a_n-L|<ε
The definition actually says quite a bit more than this, in part about at what point in the sequence this inequality is true.
janewaybos said:

Homework Equations





The Attempt at a Solution



As I understand it:

|(1/2)^n-0|<ε

|(1/2)|^n<ε
You don't need the absolute values, since 1/2 and (1/2)n are positive for all positive integers n.
janewaybos said:
then I need to solve for n?

n>(ln(ε))/ln(|1/2|)
Let's back up a bit.
You want to find a number N so that for a given ε > 0 and any n >= N, (1/2)n < ε.
Take ln of both sides: n ln(1/2) < ln(ε)
Divide both sides by ln(1/2), which is a negative number.
n > ln(ε)/ln(1/2)

The direction of the inequality changed because we divided by a negative number, ln(ε).

Note that ε is typically a very small (i.e., much less than 1), but positive number, so ln(ε) < 0, which means that ln(ε)/ln(1/2) > 0. The smaller ε is, the larger this expression is.

Take N to be the next largest integer that is greater than ln(ε)/ln(1/2). Then for any number n >= N, (1/2)n < ε.

To see how this works it might be helpful to actually pick a number for ε, say ε = 0.01. Go through the same process as above to find an index N for which all of the terms in the sequence {(1/2)n} are smaller than ε.





janewaybos said:
Then I choose N=(ln(ε))/ln(|1/2|) but I don't understand why.

n>N>(ln(ε))/ln(|1/2|)??

Given that ε>0 and n>N

then n>(ln(ε))/ln(|1/2|)

then solving for ε I get the statement

|(1/2)|^n<ε

from above. Thanks!
 

1. What is the definition of convergence?

The definition of convergence in mathematics is the property of a sequence or series where the terms get closer and closer to a specific value, known as the limit, as the number of terms increases.

2. How do you prove that the limit of (1/2)^n is equal to 0?

To prove that the limit of (1/2)^n is equal to 0, we can use the definition of convergence. This means showing that as n approaches infinity, the terms of the sequence (1/2)^n get closer and closer to 0, the limit.

3. What is the value of n in the sequence (1/2)^n?

n is the index or position of the term in the sequence. It starts at 1 and increases by 1 for each subsequent term. For example, in the sequence (1/2)^n, the first term has n=1, the second term has n=2, and so on.

4. Can you explain how the limit of (1/2)^n is related to the value 0?

As n approaches infinity, the terms of the sequence (1/2)^n get closer and closer to 0. This means that the limit of the sequence is equal to 0, as all the terms in the sequence will eventually be infinitesimally close to 0.

5. Are there any other methods to prove the convergence of a sequence?

Yes, there are other methods such as the comparison test, the ratio test, and the root test. These methods can be used to determine the convergence or divergence of a sequence, but the definition of convergence is the most fundamental and commonly used method.

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