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February 01, 2026, 01:39:15 am
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Author Topic: TT's Maths Thread  (Read 157299 times)  Share 

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TrueTears

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Re: TT's Maths Thread
« Reply #855 on: February 06, 2010, 10:31:24 pm »
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kamil check this out:

Find the sum of

This is definitely convergent by the Monotonic squeeze theorem. So...
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kamil9876

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Re: TT's Maths Thread
« Reply #856 on: February 06, 2010, 10:32:32 pm »
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Google "Basel Problem".
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Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

TrueTears

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Re: TT's Maths Thread
« Reply #857 on: February 06, 2010, 10:36:39 pm »
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haven't learnt taylor expansion of sin yet :)
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kamil9876

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Re: TT's Maths Thread
« Reply #858 on: February 06, 2010, 11:15:00 pm »
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aww
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TrueTears

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Re: TT's Maths Thread
« Reply #859 on: February 07, 2010, 07:26:49 pm »
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Hmm.. bit stuck on how to prove this limit comparison test :(

Suppose and are series with positive terms and is divergent, prove that if then is also divergent.
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kamil9876

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Re: TT's Maths Thread
« Reply #860 on: February 07, 2010, 07:36:07 pm »
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implies that there exists some natural number such that:



Can you do the rest from here? The following lemma finishes it off nicely: if then we can start from a latter term, say the term, and still get a diverging series ie:
« Last Edit: February 07, 2010, 07:40:32 pm by kamil9876 »
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Re: TT's Maths Thread
« Reply #861 on: February 07, 2010, 08:54:05 pm »
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Oh yeah thanks kamilz, you basically finished it off lolz
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TrueTears

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Re: TT's Maths Thread
« Reply #862 on: February 07, 2010, 11:10:58 pm »
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Is the series converging or diverging?

So what test do I use here... how to compare? I can see it's something along the lines of
« Last Edit: February 07, 2010, 11:39:58 pm by TrueTears »
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kamil9876

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Re: TT's Maths Thread
« Reply #863 on: February 07, 2010, 11:18:58 pm »
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yeah integral test can easily be used here.

Don't let the put you off as you can simply use:



as an upper bound for the series.
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TrueTears

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Re: TT's Maths Thread
« Reply #864 on: February 07, 2010, 11:40:27 pm »
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haha you got me, that (-1)^n i didn't know what to do with it lolz

and yeah don't really need integral test do we? it's a p series with p>1 so it's converging?
« Last Edit: February 07, 2010, 11:45:00 pm by TrueTears »
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kamil9876

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Re: TT's Maths Thread
« Reply #865 on: February 07, 2010, 11:48:01 pm »
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yeah true, p series. You can prove it with integral test but I guess if you are already familiair with it just mention p series.
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TrueTears

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Re: TT's Maths Thread
« Reply #866 on: February 07, 2010, 11:50:20 pm »
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sure, thanks heaps kamilz
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TrueTears

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Re: TT's Maths Thread
« Reply #867 on: February 08, 2010, 12:57:50 am »
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(Sorry for the trivial-like questions... it's the first time I've been exposed to these tests xD)



Is this the right way to show the series is converging.



is converging since it's a geometric series with

which the latter is converging thus is also converging.

Thus is converging...
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kamil9876

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Re: TT's Maths Thread
« Reply #868 on: February 08, 2010, 01:01:00 am »
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yep, that's correct (sum of converging series is a converging series).

Another way:


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TrueTears

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Re: TT's Maths Thread
« Reply #869 on: February 08, 2010, 01:02:02 am »
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Quote from: kamil9876 on February 08, 2010, 01:01:00 am
yep, that's correct (sum of converging series is a converging series).

Another way:



thanks bro, but how did you get the "another way", it doesn't seem obvious to me :(

edit: oh i see your edited version, lol i was like wdf, how can you bring the 4^n outside hahaha
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