maths 1B last minute questions

[Drsoccerball]

 

[InteGrand]

 

[Drsoccerball]

We roll a die successively until we roll six consecutive 6's, in which case we stop.
a) What is the probability that we stop after the thirteenth throw?
b) What is the probability that we roll the die at least 10 times?

 

[Drsoccerball]

Thanks guys just a question from the exam:





I said the first one diverges by comparison test and by Leibniz test the second one conditionally converges. Am I right?

 

[seanieg89]

Thanks guys just a question from the exam:





I said the first one diverges by comparison test and by Leibniz test the second one conditionally converges. Am I right?

Yes, this is correct. What did you compare the first series to though?

 

[Drsoccerball]

Yes, this is correct. What did you compare the first series to though?

I had like a few seconds so i just checked it with the harmonic series without checking the inequality sign... R.I.P

 

[Drsoccerball]

Okay it appears I may have read the question wrong :

 

[InteGrand]

I had like a few seconds so i just checked it with the harmonic series without checking the inequality sign... R.I.P

Here's the Wikipedia page for the divergence of the sum of reciprocals of primes (which has a few proofs of it there), if you (or anyone else) want(s) to see it: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes .

Did they get you to derive some bounds on p_n or something? It's not exactly a trivial thing to prove, that divergence.

 

[Drsoccerball]

Here's the Wikipedia page for the divergence of the sum of reciprocals of primes (which has a few proofs of it there), if you (or anyone else) want(s) to see it: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes .

Did they get you to derive some bounds on p_n or something? It's not exactly a trivial thing to prove, that divergence.

Wrote the question wrong ahah :'(

 

[seanieg89]

Using p_n for pi(n) is horrific notation, but it remains true that the first one diverges and the second one converges, and it remains true that proving divergence is nontrivial (quite similar though).

 

[InteGrand]

Using p_n for pi(n) is horrific notation, but it remains true that the first one diverges and the second one converges, and it remains true that proving divergence is nontrivial (quite similar though).

Hahaha was it π(n) that the Q. was actually referring to?

Edit: oh, just noticed Drsoccerball's post above. (Wasn't able to see the LaTeX before, I think it's playing up for me at the moment.)

Did the Q. provide any fact about the asymptotic nature of π(n)?

 

[Drsoccerball]

Hahaha was it π(n) that the Q. was actually referring to?

Edit: oh, just noticed Drsoccerball's post above. (Wasn't able to see the LaTeX before, I think it's playing up for me at the moment.)

Did the Q. provide any fact about the asymptotic nature of π(n)?

What I wrote was what the queation gave

 

[Paradoxica]

What I wrote was what the queation gave

https://en.wikipedia.org/wiki/Prime-counting_function

Alternatively you can deduce some nice (albeit crude) bounds for the prime counting function and use them to prove divergence.

 

[seanieg89]

Did the Q. provide any fact about the asymptotic nature of π(n)?

From the wording, they probably just wanted you do decide whether:

a) Neither of them converge.
or
b) At least one of them converges.

which tests students knowledge of the alternating series test whilst also leaving the red herring of the reciprocal prime series (less subject to elementary analysis) to catch out students who are not thinking critically.

 

[Drsoccerball]

For those who don't know I got 94 as my mark thanks guys