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2sqrt(n+1)-2sqrt(n) < 1/sqrt(n) < 2sqrt(n)-2sqrt(n-1)

How to prove this inequality?

(its our homework with only sage.) any tips? Please do not close it again as I have no other place where I can ask my question. And I have to use the software. Thank you.

For equality, I have to type = twice. Like this: == . But what about inequalities?

2sqrt(n+1)-2sqrt(n) < 1/sqrt(n) < 2sqrt(n)-2sqrt(n-1)

How to prove this inequality?

1.)

2!·4!·...·20!

1!·3!·...·19! (this is a fraction)

I need to solve this in a closed form. the question is that we need to find out if it is better to solve in its original form or we need to 'help it a little'? Now I typed down the whole exercise.

The other problem:

2./a)

1+ 1/2 + 1/3 +...+ 1/n >10 We need positive n that works for this.

2./b)

We need the smallest n. (not only the answer, we need to prove why that's the answer)

I wrote down all the questions for the exercises. The whole thing is in a sage document. We can upload only sage files to the server too. The teacher is tricky by the way. The subject is called Solving mathematics problems with sage. Sadly I cannot upload pictures because I'd need more karma.

I can try to do this in the analytical way, and type it in sage, but any help or tips would be great with the proof too.. :) I have no idea what my teacher wants really.

My native language is not english but I try my best to make you understand the problem. I'd like to: from k=1 to n+1 sum k^2

But when I try to solve it with sage, it gives me an error.

(k,n)=var('k,n') show(sum(k,k^2,1,n+1))

It has a problem with k^2. How to solve it?

Many thanks.