Induction inequality (1 Viewer)

[Joshmosh2]

Hello, I have come across this question, and don't know where to start:

- Prove that for any natural number n,
2(√(n-1)-1) < 1+ 1/√2 +...+ 1/√n < 2√n
http://imgur.com/jif1ZMR

for an inequality like this, what is the best approach?
For the k+1 step, i'm assuming bringing them all to a single inequality or possibly solving one component first?

 

[Trebla]

Hello, I have come across this question, and don't know where to start:

- Prove that for any natural number n,
2(√(n-1)-1) < 1+ 1/√2 +...+ 1/√n < 2√n
http://imgur.com/jif1ZMR

for an inequality like this, what is the best approach?
For the k+1 step, i'm assuming bringing them all to a single inequality or possibly solving one component first?

I would suggest doing each side separately but given that it is an induction question it seems like it is asking you to handle the two sided inequality at once

 

[Carrotsticks]

Hello, I have come across this question, and don't know where to start:

- Prove that for any natural number n,
2(√(n-1)-1) < 1+ 1/√2 +...+ 1/√n < 2√n
http://imgur.com/jif1ZMR

for an inequality like this, what is the best approach?
For the k+1 step, i'm assuming bringing them all to a single inequality or possibly solving one component first?

Does it HAVE to be via Induction?

 

[Joshmosh2]

Does it HAVE to be via Induction?

Sorry, i didn't mention it. Yes.

 

[Joshmosh2]

I would suggest doing each side separately but given that it is an induction question it seems like it is asking you to handle the two sided inequality at once

At once?
So for the k+1 step, after subbing it in, what to do next?

 

[Carrotsticks]

I'll let somebody else take care of this one.

However, I'm moving this thread to Extension 2, as these problems frequently occur under the Extension 2 topic called "Harder Extension 1" (inequalities + induction).

 

[iStudent]

I did this question before. (at least for the LHS). Are you sure there's no part i involved? (you were meant to prove an inequality and use it for the induction). If you don't have it well I assume you have to move it all to one side and prove SEPARATELY that it is greater (or less than depending on which side you're starting) than 0. (and hence deriving the part i identity)

 

[Joshmosh2]

I did this question before. (at least for the LHS). Are you sure there's no part i involved? (you were meant to prove an inequality and use it for the induction). If you don't have it well I assume you have to move it all to one side and prove SEPARATELY that it is greater (or less than depending on which side you're starting) than 0. (and hence deriving the part i identity)

There is no part i involved. It is just on its own