limit of average of terms

"show that if xn converges to l, then 1/n (x1 + x2 + ... + xn) converges to l. then show that xn = n(log(n+1)-log(n)) = -1/n log(n!) + log(n+1) goes to 1". ok, the first bit here i don't know how to show it because it is not with any particular sequence. it is just an abstract 'show that this rule is true' and i dont know how to do that. i just know that it is from my notes. any guiders? the second part, i don't know what actually needs to be shown because it is all there already. the question kind of shows that already.

Re: limit of average of terms

You can show the first part by the definition of limit.

Re: limit of average of terms

Re: limit of average of terms

it is stated that it is true for 0. so i allowed can use this information without proving it.

does that affect what i should do. i dont know about any capital K .that has never been covered

Re: limit of average of terms

i found an answer to this question but it relies on the epsilon-N definition. and uses that |x1-l|/n < epsilon/n, |x2-l|/n < epsilon/n, etc, BUT isn't this only true for n > N so, you cant say that for the low terms x1, x2 etc..?

Re: limit of average of terms

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Originally Posted by

**learning** it is stated that it is true for 0. so i allowed can use this information without proving it.

does that affect what i should do.

Yes. If you can use the claim when L = 0 (I'll write capital L because lowercase l can be confused with 1), then follow thw second part of Plato's advice:

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Originally Posted by

**Plato** Then for the case

define

and apply case #1.

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Originally Posted by

**learning** i dont know about any capital K .that has never been covered

Look, stop panicking whenever you see a letter used in a way you have not seen before. What has not been covered: that a sum of n terms can be split into two parts: the sum of the first K terms and the sum of the rest n - K terms? If you don't understand Plato's reasoning, at least you could have posed a more detailed question, e.g., how exactly is made small.

Quote:

Originally Posted by

**learning** i found an answer to this question but it relies on the epsilon-N definition. and uses that |x1-l|/n < epsilon/n, |x2-l|/n < epsilon/n, etc, BUT isn't this only true for n > N so, you cant say that for the low terms x1, x2 etc..?

Yes; that's why it was suggested to split the series into terms up to some K and the rest.

OK, speaking in more detail: if you are allowed to use the claim when L = 0 and you are given a sequence that converges to an L ≠ 0, then following Plato's advice, define . Use the theorem about the limit of the sum (or difference) of sequences. Apply the claim for L = 0 to . Expand the definition of to see what the arithmetic mean of , ..., is.

Re: limit of average of terms

Re: limit of average of terms

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