Sum of finite series Watch

psanghaLFC
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#1
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Express the finite series
log_2 a represents log to the base 2 of a

1/(log_2 a)+1/(log_3 a)+1/(log_4 a)+.....+1/(log_n a)

as a quotient of logarithms to base 2

i get

1/(log_2 a)*(1+(log_2 n!/2)) is that correct?
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SimonM
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No, try n = 1

Think of the change of base formula

\log_a x = \frac{\log_b x}{\log_b a} when x = b.

The answer becomes obvious
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psanghaLFC
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Thats what i did


log_3 a=log_2 a/log_2 3

so

1/log_3 a= log_2 3/log_2 a

similarly

1/log_n a=log_2 n/log_2 a

that then becomes

1/log_2 a(1+log_2 3+log_2 4+........+log_2 n) is this now correct?
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psanghaLFC
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Erm, isn't it defined for n>=2 in which case it would become


1/log_2 a(1+log_2 1)

1/log_2 a(1+0)=1/log_2 a
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psanghaLFC
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oh and i suppose 1=log_2 2 so its

log_2 n!/log_2 a?
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SimonM
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I'll tell you what I got



\sum_{k=2}^n \frac{1}{\log_k a} = \sum_{k=2}^n \log_a k = \log_a n!

Which is \frac{\log_2 n!}{\log_2 a}
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psanghaLFC
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yes, that is the same as me, the next part asks whether it converges. I say, no because log 2 n!->inf as n->inf
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Cities
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Out of curiosity, what level / module is this?
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cadaeibfeceh
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this proves that the IGCSE I just sat was not maths but mere arithmetic...
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EierVonSatan
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(Original post by n1r4v)
Out of curiosity, what level / module is this?
well i learn't 'convergance of infinite series' at undergraduate level...the ratio test, cauchys integral test etc

*shudders*
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SimonM
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\sum_{k=2}^n \log_a k

For all k > a we have \log_a k > 1. Therefore we are adding up terms greater than one, hence it obviously diverges
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Cities
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(Original post by EierVonSatan)
well i learn't 'convergance of infinite series' at undergraduate level...the ratio test, cauchys integral test etc

*shudders*
~Ah cool... interesting that's taught in a Chemistry degree . I was wondering what applications that had in Chemistry?

Though I do remember you saying something about Jahn-Teller distortions and I looked it up, and it required an understanding of group theory, so I guess quite a bit of degree level Maths is taught in Chemistry!

-----

Although it's not a good idea to say this... NO NOT GOGSOC . That was the reason for my rep plunging from a mediocre 300 or so to -828. I also got some pretty disgusting comments from them (including a picture which looks a lot like a spotty penis:mad: ) all because I wanted to know who randomly neg repped me from that thread (I maintain I only became annoying AFTER the neg reps started to come in!). I hate them all [except you of course:p:]
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nota bene
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(Original post by n1r4v)
I was wondering what applications that had in Chemistry?
Look up the Madelung constant for something. In fact I made a thread on that not too long ago, I'll go find it; http://www.thestudentroom.co.uk/show...2#post11821872
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Cities
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(Original post by nota bene)
Look up the Madelung constant for something. In fact I made a thread on that not too long ago, I'll go find it; http://www.thestudentroom.co.uk/show...2#post11821872
Thanks for taking the time out to find it. It's interesting!
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EierVonSatan
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(Original post by n1r4v)
~Ah cool... interesting that's taught in a Chemistry degree . I was wondering what applications that had in Chemistry?

Though I do remember you saying something about Jahn-Teller distortions and I looked it up, and it required an understanding of group theory, so I guess quite a bit of degree level Maths is taught in Chemistry!
Well we only learn't it because its was in the i maths course for scientists, its probably more relevant for the physicists saying that nota bene found an example

You don't really need to know group theroy to understand Jahn-teller, just a bit of its notation (its a first year topic)...the group theroy is more useful for things like quantum mechanics, molecular symmetry and spectroscopic transistions which are significantly more advanced topics. I've attached the lecture notes for a third year course in spectroscopy if you want to take a look, but bare in mind ithis is not for the faint hearted
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Cities
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(Original post by EierVonSatan)
Well we only learn't it because its was in the i maths course for scientists, its probably more relevant for the physicists saying that nota bene found an example

You don't really need to know group theroy to understand Jahn-teller, just a bit of its notation (its a first year topic)...the group theroy is more useful for things like quantum mechanics, molecular symmetry and spectroscopic transistions which are significantly more advanced topics. I've attached the lecture notes for a third year course in spectroscopy if you want to take a look, but bare in mind ithis is not for the faint hearted
That's brilliant, thanks

Since I'm probably going to have to do a lot of medicine reading in the next few months and get really bored, this sort of stuff will be nice to read around on. I'm quite interested in the other sciences, and since I haven't found too much well-explained introductory stuff on the internet, those notes are diamond! I would have repped you if it meant anything, so instead, :adore:
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psanghaLFC
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This is from a book "The Mathematical Olympiad handbook" is basically just BMO round1 problems
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