Conversation with merc248
(2007-09-25 18:05:41) kanzure: quick, what's the power series *really* about?
(2007-09-25 18:05:45) kanzure: Wikipedia doesn't explain much of it to me
(2007-09-25 18:25:25) merc248: the power series? so, as far as i understand, it's to represent functions with polynomials
(2007-09-25 18:25:45) merc248: which is very important in analysis since you have what's called "analytic functions," which are functions that are represented by power series within a given "interval of convergence"
(2007-09-25 18:26:07) kanzure: A guy in #not-math is giving a quick lecture on irc.freenode.net on how combinatorics and the power series are related, apparently the power series can be used for counting via the coefficients
(2007-09-25 18:26:13) kanzure: though I'm not entirely sure what's going on
(2007-09-25 18:26:15) merc248: the regular functions that you can express as a single term or a finite sum are a subset of the analytic functions; there's a bunch of other ones that can ONLY be expressed as a power series
(2007-09-25 18:26:27) kanzure: "interval of convergence" ?
(2007-09-25 18:26:30) merc248: hmm, seems prudent, i don't know mucha bout combinatorics though
(2007-09-25 18:27:05) merc248: yeah, say you have a sum from 0 to inf of say... x^2
(2007-09-25 18:27:19) merc248: it's whatever values of x that makes the entire sum finite
(2007-09-25 18:27:24) merc248: so in this case
(2007-09-25 18:27:37) merc248: you'd have the absolute values of x less than 1
(2007-09-25 18:27:54) merc248: because in any other case, you'll have a divergent series. make sense?
(2007-09-25 18:28:40) merc248: so in this case, you say that the radius of convergence
(2007-09-25 18:28:41) merc248: is 1
(2007-09-25 18:28:46) kanzure: what?
(2007-09-25 18:28:47) kanzure: wait
(2007-09-25 18:28:52) kanzure: if you're taking the sum from 0 to inf of x^2
(2007-09-25 18:29:03) kanzure: how are you going to find a value of x that will make the sum finite?
(2007-09-25 18:29:23) merc248: EVEN if say if x = 1 it diverges; it's the supremum of the values that DO converge, if that makes any sense
(2007-09-25 18:29:39) merc248: there's a couple of tests i think that you can do?
(2007-09-25 18:30:12) merc248: aww fuck, hold on, i forgot what they were, but they had to do with the root test and the ratio test
(2007-09-25 18:30:41) merc248: R = reciprocal of either the root test or the ratio test on the series
(2007-09-25 18:30:42) kanzure: No, that makes no sense
(2007-09-25 18:30:45) merc248: lol
(2007-09-25 18:30:48) merc248: yeah, i suck at explaining things
(2007-09-25 18:30:48) kanzure: can you explain divergence/convergence?
(2007-09-25 18:31:05) merc248: in a few words, divergence = values shooting off into infinity
(2007-09-25 18:31:10) merc248: convergence = finite values
(2007-09-25 18:31:19) merc248: typically when you're talking about divergence/convergence
(2007-09-25 18:31:20) kanzure: so
(2007-09-25 18:31:25) kanzure: if a sum is from some finite number -> infinity
(2007-09-25 18:31:27) merc248: you're talking about an infinite sum or product or some other infinite process
(2007-09-25 18:31:29) kanzure: how can it do anything but diverge?
(2007-09-25 18:31:45) merc248: ahh, check out... 1 / x^2
(2007-09-25 18:31:53) merc248: notice that if x > 1
(2007-09-25 18:31:57) merc248: you'll have a finite sum
(2007-09-25 18:32:22) merc248: because basically, the rate at which 1 / x^2 decreases is much faster than you can sum them up
(2007-09-25 18:32:29) kanzure: really?
(2007-09-25 18:32:32) merc248: yeah
(2007-09-25 18:32:37) kanzure: 1/(x^2) decreases faster than it would increase?
(2007-09-25 18:32:37) kanzure: so:
(2007-09-25 18:33:07) kanzure: sum of x^-2 from 0 -> infinity is faster than sum of x^-2 from 1 -> 0 ?
(2007-09-25 18:33:18) merc248: now, 1 / x is DIVERGENT. if you write out the first couple of terms, you can see why; you can group up i think 2^n terms together for any n and you always get 1/2 out of it, so the first 2^1 terms = 1/2, 2^2 terms = 1/2, etc.... so basically, you add 1/2 infinite amounts of times
(2007-09-25 18:33:25) merc248: well
(2007-09-25 18:33:28) merc248: x^-2?
(2007-09-25 18:33:30) merc248: or x^2
(2007-09-25 18:33:32) kanzure: yes,
(2007-09-25 18:33:37) kanzure: x^-2 == 1/x^2
(2007-09-25 18:33:42) merc248: err, ahh right :p
(2007-09-25 18:33:45) kanzure: I was playing with your example
(2007-09-25 18:33:56) merc248: yeah, pretty much, though i think
(2007-09-25 18:34:00) merc248: at least if you're talking about an integral
(2007-09-25 18:34:06) kanzure: no, I'm talking about a summation
(2007-09-25 18:34:06) kanzure: heh'
(2007-09-25 18:34:06) merc248: x^-2 would diverge from 1 -> 0
(2007-09-25 18:34:21) kanzure: well, integrals involve summations, yes
(2007-09-25 18:34:26) merc248: well yeah, though there's a thing called the "integral test" which ties the infinite series to its corresponding integral
(2007-09-25 18:34:42) merc248: basically, if the integral converges, the series converges
(2007-09-25 18:34:51) kanzure: so what would an integral 'converging' look like?
(2007-09-25 18:35:01) kanzure: I keep thinking of terms of limits approaching a value or not approaching anything
(2007-09-25 18:35:05) kanzure: I am not sure if that's the right way to think abotu this
(2007-09-25 18:35:05) kanzure: *about
(2007-09-25 18:35:19) merc248: ahh, okay, so if you have an integral from 0 to infinity of x^-2
(2007-09-25 18:35:24) kanzure: hmm
(2007-09-25 18:35:25) merc248: you have to have a limiting process
(2007-09-25 18:35:30) merc248: on the infinity part of the integral
(2007-09-25 18:35:32) kanzure: I have not evaluated any integrals from finite numbers to infinite numbers
(2007-09-25 18:35:36) kanzure: so ...
(2007-09-25 18:35:38) merc248: hmm
(2007-09-25 18:35:44) kanzure: maybe you can explain this as well?
(2007-09-25 18:35:44) kanzure: haha
(2007-09-25 18:35:57) merc248: well, bad example before, let's say integral from 1 to infinite of x^-2
(2007-09-25 18:36:11) merc248: you have an improper integral with infinity being the part where it fucks up
(2007-09-25 18:36:21) merc248: basically, you replace infinity with a variable T
(2007-09-25 18:36:23) merc248: and say
(2007-09-25 18:36:31) kanzure: improper integrals ... hmm. How do you work with improper integrals?
(2007-09-25 18:36:33) merc248: lim T-> inf of the integral from 1 -> T
(2007-09-25 18:36:39) kanzure: ah
(2007-09-25 18:36:47) merc248: then you evaluate it like you would a regular integral
(2007-09-25 18:36:58) kanzure: that's useful
(2007-09-25 18:37:10) kanzure: then you have lim T->inf [of your evaluated integral from 1->T] ?
(2007-09-25 18:37:12) merc248: (NOTE! SUBSTITUTE IN T FOR INFINITY AS SOON AS YOU CAN BEFORE FOOLING AROUND WITH THE INTEGRAL)
(2007-09-25 18:37:21) merc248: (I LOST MANY POINTS ON A TEST BECAUSE I DIDN'T DO THAT :O)
(2007-09-25 18:37:26) kanzure: haha
(2007-09-25 18:37:33) merc248: yep, pretty much
(2007-09-25 18:37:34) kanzure: I lost many points on a test because I failed to learn trig substitution ;)
(2007-09-25 18:37:38) merc248: haha
(2007-09-25 18:37:52) merc248: so like, let's just say, integral from 1 -> T for now
(2007-09-25 18:37:53) kanzure: don't know my derivatives of inverse trigonometric functions well enough :(
(2007-09-25 18:37:54) merc248: of 1/x^2
(2007-09-25 18:37:54) kanzure: alright
(2007-09-25 18:38:17) merc248: then you get something like -x^-1 as your integral
(2007-09-25 18:38:20) merc248: evaluated from 1 to T
(2007-09-25 18:38:39) merc248: -1/T + 1
(2007-09-25 18:38:48) merc248: now, you do the limiting process: T -> infinite
(2007-09-25 18:38:50) merc248: err, infinity
(2007-09-25 18:38:55) merc248: -1/T -> 0
(2007-09-25 18:38:58) merc248: so you're left with 1
(2007-09-25 18:39:10) merc248: basically
(2007-09-25 18:39:18) kanzure: huh?
(2007-09-25 18:39:19) kanzure: what just happened?
(2007-09-25 18:39:23) merc248: hahaha
(2007-09-25 18:39:37) merc248: alright, so basically, we got our integral, which is -x^-1
(2007-09-25 18:39:41) kanzure: yes
(2007-09-25 18:39:44) merc248: from x^-2
(2007-09-25 18:39:48) merc248: then we evaluate from 1 to T
(2007-09-25 18:39:53) merc248: so we get -1/T + 1
(2007-09-25 18:40:02) kanzure: ookay
(2007-09-25 18:40:03) kanzure: *okay
(2007-09-25 18:40:11) merc248: and then, we bring in our limiting process: lim T-> infinity of -1/T + 1
(2007-09-25 18:40:24) merc248: T-> infinity implies that -1/T -> 0
(2007-09-25 18:40:38) merc248: so we're left with just our + 1 term
(2007-09-25 18:41:03) kanzure: yes
(2007-09-25 18:41:31) merc248: to tie that in to the whole integral test thing i was talking about, since the sum from 1 to infinity of x^-2 can be tested against the integral from 1 to infinity of x^-2
(2007-09-25 18:41:36) merc248: and since the integral has a finite value
(2007-09-25 18:41:42) merc248: the series has a finite value, therefore convergent
(2007-09-25 18:42:27) kanzure: so
(2007-09-25 18:42:32) kanzure: there are cases where the integral does not have a finite value?
(2007-09-25 18:42:38) merc248: oh, definitely
(2007-09-25 18:42:46) merc248: consider the integral from 0 to 1 of x^-2
(2007-09-25 18:42:53) kanzure: and if the integral does not have a finite value, can the series have an infinite value?
(2007-09-25 18:43:02) merc248: if you just draw the graph of x^-2, you can see that the left part from 0 to 1 has infinite area
(2007-09-25 18:43:07) kanzure: I mean, does the value of the integral determine the value of the series?
(2007-09-25 18:43:13) kanzure: (finiteness/infiniteness)
(2007-09-25 18:43:13) merc248: it doesn't determine the exact value
(2007-09-25 18:43:18) merc248: err
(2007-09-25 18:43:20) merc248: hmm
(2007-09-25 18:43:22) kanzure: but if one's finite, is the other necessarily finite?
(2007-09-25 18:43:32) merc248: that's a good question, i don't know if it goes both ways necessarily, let me check
(2007-09-25 18:43:38) kanzure: alright :)
(2007-09-25 18:44:01) merc248: ahh, yeah
(2007-09-25 18:44:02) merc248: it does
(2007-09-25 18:44:08) merc248: so if the integral diverges, the series diverges
(2007-09-25 18:44:21) kanzure: that's useful
(2007-09-25 18:44:49) kanzure: and in this specific instance, the term 'diverges' means that the integral does not approach a finite value
(2007-09-25 18:45:07) merc248: yep
(2007-09-25 18:45:22) merc248: in all cases, divergent means an infinite process does NOT reach a finite value
(2007-09-25 18:45:24) merc248: err
(2007-09-25 18:45:40) merc248: this might confuse you a bit, so kinda put it aside separate from what i just told you
(2007-09-25 18:45:47) merc248: (ie, don't mix it into the previous stuff i said)
(2007-09-25 18:45:52) merc248: but there's a way to sum divergent series
(2007-09-25 18:45:56) merc248: such that it has a finite value
(2007-09-25 18:46:01) merc248: check out cesaro sums on wikipedia
(2007-09-25 18:46:20) merc248: you can use cesaro sums to make 1 - 1 + 1 - 1 + 1... approach a finite value, that is, 1/2
(2007-09-25 18:46:28) kanzure: http://en.wikipedia.org/wiki/Cesàro_summation ?
(2007-09-25 18:46:35) kanzure: hahah
(2007-09-25 18:46:37) kanzure: that's wacky :)
(2007-09-25 18:46:47) merc248: (oh shit, okay, sorry, i should say that divergent means it is not convergent; convergence means it approaches a single finite value)
(2007-09-25 18:47:04) merc248: (in the last example, 1 - 1 + 1 - 1... definitely does not shoot off to infinity, but it doesn't converge to a single value)
(2007-09-25 18:47:14) merc248: yep :)
(2007-09-25 18:47:21) kanzure: == Definition ==
Let {''a''<sub>n</sub>} be a [[sequence]], and let
:<math>s_k = a_1 + \cdots + a_k</math>
be the ''k''th partial sum of the series
:<math>\sum_{n=1}^\infty a_n</math>.
The sequence {''a''<sub>n</sub>} is called '''Cesàro summable''', with Cesàro sum α, if
:<math>\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = \alpha</math>.
(2007-09-25 18:48:19) merc248: ahhhh
(2007-09-25 18:48:25) kanzure: hard to read, but :)
(2007-09-25 18:48:27) merc248: i can't read fucking latex :O
(2007-09-25 18:48:31) merc248: at least not yet
(2007-09-25 18:48:32) kanzure: try typing it out in class
(2007-09-25 18:48:33) kanzure: haha
(2007-09-25 18:48:35) merc248: lol
(2007-09-25 18:48:38) kanzure: that'll be a quick way to learn to read it
(2007-09-25 18:48:43) merc248: oh man
(2007-09-25 18:48:43) kanzure: it doesn't render nicely with the eyes
(2007-09-25 18:48:49) kanzure: I've been trying to figure out a method to render it in nearly-real time
(2007-09-25 18:48:54) kanzure: so that maybe when I press enter it renders the last line
(2007-09-25 18:48:54) merc248: lol, good luck
(2007-09-25 18:48:55) kanzure: wouldn't that be nice?
(2007-09-25 18:49:02) merc248: best way to render it in real time: get a notepad, get pencil
(2007-09-25 18:49:03) kanzure: There's a way to do with emacs, there's this special "preview-latex" option
(2007-09-25 18:49:14) merc248: hmm
(2007-09-25 18:49:16) kanzure: but my Debian package manager is broken at the moment (have to fix that when I get some free time)
(2007-09-25 18:49:21) merc248: ahh shitty
(2007-09-25 18:49:24) kanzure: btw,
(2007-09-25 18:49:36) kanzure: http://heybryan.org/school/Calculus/index2.html
(2007-09-25 18:49:39) kanzure: My cal-2 class :)
(2007-09-25 18:50:43) merc248: aww yeah
(2007-09-25 18:50:44) merc248: rofl
(2007-09-25 18:51:12) merc248: dude, i gotta say btw, i appreciate you talking about math with me especially today
(2007-09-25 18:51:21) merc248: i was getting especially rusty over the past couple of weeks and have been lazy in reading math
(2007-09-25 18:51:30) merc248: and since i got fucking class starting tomorrow, i need all the refreshment as i can get
(2007-09-25 18:51:30) merc248: :O
(2007-09-25 18:51:52) kanzure: I am glad to have somebody to talk with in the first place
(2007-09-25 18:52:12) merc248: haha
(2007-09-25 18:52:18) merc248: man
(2007-09-25 18:52:24) kanzure: yeah, so in class right now
(2007-09-25 18:52:35) kanzure: we're going over limited exponential growth
(2007-09-25 18:52:55) merc248: aye
(2007-09-25 18:53:04) kanzure: where we solve a separable differential equation in order to find the exponentiated integrated form
(2007-09-25 18:53:34) kanzure: probably something to do with series, but we haven't been introduced to them "Formally"
(2007-09-25 18:55:11) merc248: ahhh right
(2007-09-25 18:55:36) merc248: you can generally solve ordinary diffeq's with series i think
(2007-09-25 18:55:39) merc248: if it's linear
(2007-09-25 18:55:45) kanzure: what do you mean?
(2007-09-25 18:55:50) kanzure: ODEs I know about
(2007-09-25 18:55:55) kanzure: but I am not sure of the connection with series
(2007-09-25 18:56:01) kanzure: I thought that you just simply solve ODEs through typical first-year methods
(2007-09-25 18:56:15) merc248: basically you ASSUME that you can solve it with power series; ie, summation 0-> inf of x^n
(2007-09-25 18:56:27) merc248: so you substitute that in for the y's in your equation
(2007-09-25 18:56:34) merc248: and do derivatives, etc.
(2007-09-25 18:56:46) merc248: then you do a bunch of summation end point manipulation, etc. etc.
(2007-09-25 18:57:20) merc248: that is, you want to make it so your sums all match up on the bottom end, so if you have sum 1-> inf, sum 0-> inf, etc., you want the second one to go from 1->inf so you can combine the two series together
(2007-09-25 18:58:07) merc248: (now, to do that, all you do is you take the 0th term out, like say if you have the sum 0->inf of a_n, it would be equal to a_0 + sum 1->inf of a_n)
(2007-09-25 18:58:45) merc248: now that you have, for example, sum 1-> inf a_n + sum 1-> inf b_n, you can combine the two into a single series: sum 1->inf a_n + b_n
(2007-09-25 18:59:19) merc248: and i _THINK_ what you do next is that you notice what values the coefficients have to be at in order to get whatever's now on the right side of the equation
(2007-09-25 18:59:27) merc248: (this is probably confusing the fuck out of you, so i'll stop :p)
(2007-09-25 19:00:09) kanzure: (sum from a->b of x) + (sum from a->b of y) == sum from a->b of (x + y) ?
(2007-09-25 19:00:14) merc248: yep
(2007-09-25 19:00:49) merc248: now, under the context of diffeq's, you'll of course have a bunch of stuff where you'll have sum 0->inf x^n, sum 1->inf nx^(n-1), etc.
(2007-09-25 19:00:54) kanzure: I need to do some quick power browsing
(2007-09-25 19:01:15) merc248: so if you had sum 0->inf x^n + sum 1-> inf nx^(n-1), what would you do to get them under a single power series?
(2007-09-25 19:01:27) merc248: or i should say, a single summand
(2007-09-25 19:01:46) kanzure: http://en.wikipedia.org/wiki/Integral_test_for_convergence
(2007-09-25 19:01:58) kanzure: http://tutorial.math.lamar.edu/classes/calcII/IntegralTest.aspx
(2007-09-25 19:02:09) kanzure: http://jtaylor1142001.net/calcjat/Solutions/Series/IntTest/IntTest1/IntTest1Layers.htm (an online test?)
(2007-09-25 19:02:30) kanzure: http://pirate.shu.edu/projects/reals/numser/tests.html A list of convergence tests for series
(2007-09-25 19:02:46) merc248: ohhh man, good stuff
(2007-09-25 19:02:48) kanzure: http://www.sosmath.com/calculus/improper/series/series.html Short page on improper integrals
(2007-09-25 19:03:04) kanzure: What's a p-series?
(2007-09-25 19:03:13) merc248: you'll most likely learn comparison test, lim comparison, geometric series test, p series, root, ratio, alternating, and integral
(2007-09-25 19:03:21) merc248: i think it's x^p
(2007-09-25 19:03:22) merc248: i believe
(2007-09-25 19:03:29) kanzure: http://www.math.unh.edu/~jjp/radius/radius.html Excellent lecture on the convergence of infinite series
(2007-09-25 19:03:38) merc248: err wait, n^-p
(2007-09-25 19:03:49) merc248: aye
(2007-09-25 19:04:26) kanzure: http://planetmath.org/encyclopedia/IntegralTest.html
(2007-09-25 19:04:29) kanzure: don't know if that's any good
(2007-09-25 19:04:32) kanzure: looks kind of boring, but formal