the series a_n doesn't have a limit.
prove that the series (1/n) doesnt go to 1.
use the definiton of lim:
for every epsilon>0......
To prove that the series (1/n) does not converge to 1, we will use the definition of a limit.
Definition: The series {a_n} converges to a real number L if, for every ε > 0, there exists a positive integer N such that for all n ≥ N, |a_n - L| < ε.
Now, let's assume that the series (1/n) does converge to 1.
According to the definition, for any ε > 0, there should exist a positive integer N such that for all n ≥ N, |1/n - 1| < ε.
Let's consider ε = 1/2. According to the definition, there exists a positive integer N such that for all n ≥ N, |1/n - 1| < 1/2.
Now, let's choose any positive integer n ≥ N. We have:
|1/n - 1| < 1/2
Adding 1 to both sides:
|1 - 1/n| < 1/2
Rearranging the absolute value:
-1/2 < 1 - 1/n < 1/2
Subtracting 1 from all parts:
-3/2 < -1/n < -1/2
Taking the reciprocal:
-2/3 > n > -2
This is a contradiction since n should be a positive integer, but we end up with n being less than -2. Hence, our assumption that (1/n) converges to 1 is false.
Therefore, we have proven that the series (1/n) does not converge to 1.