- #1
Dick said:You can't pick an N such that x_n<epsilon/n. epsilon/n isn't a constant. You didn't use that x_n is decreasing... Where does your proof fail? Now think how that can't happen if the series is decreasing.
Abraham said:So I went about thinking, and here's my new proof. Thanks for the help!
Dick said:I don't see any contradiction there. Where is it? First you say n_i+1<n_i, then you say n_i>n_i+1. That looks like the say thing to me. The proof you want to use as a model is the most common proof the harmonic series diverges. Look at the first proof in http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
Abraham said:Sorry, I incorrectly typed this into Latex. I meant to say that the sequence n_i is a strictly increasing sequence. It is the sequence of all n for which n*x_n is greater or equal to epsilon. Progressing along the natural numbers, n(1) must be less than n(2), and n(i) < n(i+1). The contradiction arises when the decreasing x_n sequence: x(i) > x(i+1), implies that n(i) > n(i+1).
That is, 1/n(i) > 1/n(i+1) implies n(i) < n(i+1), a contradiction.
My rationale for doing it this way instead of a modified harmonic series proof, is that don't know any values of the n_i sequence. I feel as though I need actual numbers to show that a series is unbounded. The harmonic series proof of divergence uses the fact that we can group terms in such a way that we have a constant unbounded sequence of 1/2.
I am guaranteed the existence of n's from the statement, "there exist n such that n*x_n is greater or equal to epsilon". But I gain no numerical information to show that the series of those n is unbounded. What do you think of this?
I will upload a corrected PDF soon
Abraham said:Ah, I was so convinced I could do something clever using just the ordering. Also, if you have the time to explain, why was the previous proof incorrect? I arrived at a contradiction. I see that I didn't make use of all the hypotheses; I'm wondering if that makes it an insufficient contradiction?
Anyways, many thanks for the hints.
-A.
The limit of a function is the value that a function approaches as the input variable gets closer and closer to a specific value. It is used to describe the behavior of a function at a particular point and is an essential concept in calculus and analysis.
A limit is defined using the notation of lim as follows: limx→a f(x) = L, where a is the point at which the function is being evaluated, f(x) is the function, and L is the limit value.
If a limit equals 0, it means that the function is approaching 0 as the input variable gets closer to a specific value. This indicates that the function is getting smaller and smaller, but never reaches 0 at that particular point.
To prove that a limit equals 0, you need to show that the function value can be made arbitrarily close to 0 by choosing appropriate values for the input variable. This can be done using the epsilon-delta definition of a limit or by using algebraic manipulation and theorems of limits.
Understanding limits is crucial in many areas of science and engineering. It is used in physics to calculate the instantaneous velocity and acceleration of an object, in economics to determine the marginal cost and revenue of a product, and in computer science to analyze algorithms and optimize their performance.