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!