If <munder> &#x2211;<!-- ∑ --> n </munder> <mo fence="false" stretchy="false">&#x2016

kolutastmr

kolutastmr

Answered question

2022-07-16

If n x n x n 1 2 < , is ( x n ) Cauchy?

Answer & Explanation

Jayvion Tyler

Jayvion Tyler

Beginner2022-07-17Added 23 answers

Yes. Let ε > 0 and find n 0 N such that j = n 0 + 1 h j 2 < ε 2 . Now let m > n n 0 . We have x m x n = x m x m 1 + x m 1 x m 2 + + x n + 1 x n = j = n + 1 m h j . Therefore
x n x m 2 = j = n + 1 m h j 2 = j = n + 1 m h j 2 j = n 0 + 1 h j 2 < ε 2
so x n x m < ε

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?