8. *Suppose that ∑ ak converges. Prove that limn...

Transcript

00:01 In this problem, we're told that the limit as n goes to infinity of some sequence of numbers zn is equal to a.

00:16 And we're asked to prove that the limit of n goes to infinity of the following, 1 over n times z1 plus z2 plus dot dot dot plus zn is also equal to a.

00:51 Well, the first thing i'm going to do is i'm going to rewrite this in a nicer form using summation notation.

00:57 So this is the limit as n goes to infinity of 1 over n, the sum from i equals 1 to n of z i.

01:19 And now what we want to do is show that this limit is equal to a.

01:24 So the first thing i'm going to do is recall what the limit function, limit means in terms of an epsilon delta proof.

01:32 This limit at the top means that for all epsilon greater than zero there exists a capital n which is in the the integers such that the difference between zn and a is less than epsilon for all n greater than this capital n.

02:06 So this is the definition of a limit.

02:08 This means that there exists some n such that after that every single point in the sequence gets closer to the limiting limiting value.

02:17 And now we want to use this to prove that the limit of this sequence is a.

02:24 And so what we need to do is we need to show that there exists an n, a capital n, for any epsilon that the difference between this and a is less than any given epsilon.

02:41 One.

02:41 So let us consider this first...

Question

8. Suppose that $\sum a_k$ converges. Prove that \begin{equation} \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} ka_k = 0. \end{equation*}

Question

8. *Suppose that $\sum a_k$ converges. Prove that \begin{equation*} \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} ka_k = 0. \end{equation*}

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8. Suppose that $\sum a_k$ converges. Prove that \begin{equation} \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} ka_k = 0. \end{equation*}