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convergence and natural numbers

First attachment shows my question regarding convergence

s
econd attachment shows my quick question about natural numbers

Original post by ChocInABox
First attachment shows my question regarding convergence

s
econd attachment shows my quick question about natural numbers


The sum of the first n natural numbers is written as Σr=1nr \Sigma_{r=1}^n r
Original post by ChocInABox
First attachment shows my question regarding convergence

s
econd attachment shows my quick question about natural numbers


The most common formal test for convergence is D'Alembert's Ratio Test:
If Σur is a series of positive terms and limnpun+1un=p then \Sigma u_r \text{ is a series of positive terms and }\lim _{n \rightarrow p} \frac{u_{n+1}}{u_n}=p \text{ then}
 if p<1   Σur converges  \text{ if }p <1 \ \ \ \Sigma u_r \text{ converges }
 if p>1   Σur diverges  \text{ if }p >1 \ \ \ \Sigma u_r \text{ diverges }
If p=1 the test is inconclusive.
Reply 3
Original post by brianeverit
The most common formal test for convergence is D'Alembert's Ratio Test:
If Σur is a series of positive terms and limnpun+1un=p then \Sigma u_r \text{ is a series of positive terms and }\lim _{n \rightarrow p} \frac{u_{n+1}}{u_n}=p \text{ then}
 if p<1   Σur converges  \text{ if }p <1 \ \ \ \Sigma u_r \text{ converges }
 if p>1   Σur diverges  \text{ if }p >1 \ \ \ \Sigma u_r \text{ diverges }
If p=1 the test is inconclusive.


Many Thanks for the responses :h::

would you say this second bit is expected at AS level - I've never been taught this test before at school LOOL.

Original post by ChocInABox
Many Thanks for the responses :h::

would you say this second bit is expected at AS level - I've never been taught this test before at school LOOL.


Not sure but it is certainly how we establish the conditions for convergence of most of the common power series such as ln(1+x),sinx,cosx,ex \ln(1+x), \sin x, \cos x, e^x etc.
Reply 5
Original post by brianeverit



Not sure but it is certainly how we establish the conditions for convergence of most of the common power series such as ln(1+x),sinx,cosx,ex \ln(1+x), \sin x, \cos x, e^x etc.


Oh okay :smile:

Just wondering, in what you wrote does the un stand for any term of the sequence?
Original post by ChocInABox
Many Thanks for the responses :h::

would you say this second bit is expected at AS level - I've never been taught this test before at school LOOL.

This kind of test was never taught to me even at A2. The syllabus may, of course, have changed in the two years since.
Reply 7
Original post by ChocInABox
Many Thanks for the responses :h::

would you say this second bit is expected at AS level - I've never been taught this test before at school LOOL.


Where did this 'spec' come from that is referred to in your attachments?

Formal tests for convergence aren't taught for A level as far as I'm aware, with the possible exception of the OCR MEI board - they sometimes look at quite advanced concepts as part of FM.
Original post by ChocInABox
Oh okay :smile:

Just wondering, in what you wrote does the un stand for any term of the sequence?


Yes un u_n is the nth term of a sequence.
Reply 9
Original post by davros
Where did this 'spec' come from that is referred to in your attachments?

Formal tests for convergence aren't taught for A level as far as I'm aware, with the possible exception of the OCR MEI board - they sometimes look at quite advanced concepts as part of FM.



its OCR MEI lol oh gosh I just took another look at it now and I've realised that was in the heading saying 'Exclusions' ohh dear :colondollar: so it appears I do not need to know it ..ooops

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