Extension 2 limiting sum problem with a sequnce (2 Viewers)

[Drsoccerball]

 

[InteGrand]

u_{n+1} = u_n +u_n ^2 and u_1 = 1/3
Find the sum from n=1 to infty of 1/(1+u_n) Latex dont work no more

I think LaTeX started working again:

 

[Drsoccerball]

I think LaTeX started working again:

Let me fix up my post then

 

[Drsoccerball]

I got to :

But didnt know what to do from there tried everything...

 

[braintic]

You might want to fix up your k's and n's.

 

[Drsoccerball]

You might want to fix up your k's and n's.

Fixed.

 

[kawaiipotato]

Where did you get this question from?

 

[seanieg89]

 

[seanieg89]

Strictly speaking, one would have to justify the above with a proof that the sum is a convergent one, but I doubt this would be required in the HSC.

This could be done (for example) by comparison to a geometric series.

 

[Drsoccerball]

Howd you get the second step?

 

[InteGrand]

Howd you get the second step?

 

[seanieg89]

It's clear that the sequence is increasing because

This means that for all k.

Hence



So each term in our positive series is bounded by the corresponding term in the geometric series with initial term 1/3 and ratio 3/4. We know this geometric series is convergent, and so S must be as well. (*)

Ps you really should try to prove these things yourself rather than asking someone else straight away, this is the best way to improve as a mathematician.





(*) It is one of the first theorems you prove in real analysis that a monotonic bounded sequence is convergent. This predictable fact is as low level as you can reduce a proof of a question like this to in MX2. Proving this statement itself requires a rigorous definition of the reals.