GodzillaDeuce Posted April 3, 2012 Share Posted April 3, 2012 2. First few terms look like: (-1)+(-1/2)+(-1/3)+(-1/4)+(-1/5)+(-1/6)+(-1/7)+1/8+1/9+1/10+1/11+1/12+...+1/26+(-1/27)+...+(-1/63)+1/64+... what's happening is every time you hit a cubic number (8,27,64,125,...) the terms will switch sign until the next cubic number. so let a1=1+1/2+1/3+1/4+1/5+1/6+1/7 a2=1/8+1/9+...+1/26 a3=1/27+...1/63 etc your series is then sum((-1)nan) if an approaches zero (which it does) then the alternating series must converge (alternating series test) to show the an's converge to zero: a1<7 a2<19/8 a3<37/27 a4<61/64 . . . an<(3n2+3n+1)/n3 which goes to zero. Done. FYI the numerators are called centred hexagonal numbers Link to comment Share on other sites More sharing options...
rgrewal3 Posted April 3, 2012 Author Share Posted April 3, 2012 2. First few terms look like: (-1)+(-1/2)+(-1/3)+(-1/4)+(-1/5)+(-1/6)+(-1/7)+1/8+1/9+1/10+1/11+1/12+...+1/26+(-1/27)+...+(-1/63)+1/64+... what's happening is every time you hit a cubic number (8,27,64,125,...) the terms will switch sign until the next cubic number. so let a1=1+1/2+1/3+1/4+1/5+1/6+1/7 a2=1/8+1/9+...+1/26 a3=1/27+...1/63 etc your series is then sum((-1)nan) if an approaches zero (which it does) then the alternating series must converge (alternating series test) to show the an's converge to zero: a1<7 a2<19/8 a3<37/27 a4<61/64 . . . an<(3n2+3n+1)/n3 which goes to zero. Done. FYI the numerators are called centred hexagonal numbers Link to comment Share on other sites More sharing options...
GodzillaDeuce Posted April 3, 2012 Share Posted April 3, 2012 3. first term is positive, next 3 negative, next 3 positive, next 3 negative, next 4 positive, etc. the terms of your series can actually be written as (-1)floor(n/pi+.5)|cos(n)|/sqrt(n) the exponent floor(n/pi+.5) basically counts how many integers are between zeros of the cosine function. so you could leave the first term alone, group the next three negatives together, then the three positives, etc., just as in 2. it gets a little complicated keeping track of where the "4"s are, but you can see that it will converge quite easily Link to comment Share on other sites More sharing options...
GodzillaDeuce Posted April 3, 2012 Share Posted April 3, 2012 i'm still stumped by 1. actually LOL Link to comment Share on other sites More sharing options...
rgrewal3 Posted April 3, 2012 Author Share Posted April 3, 2012 i'm still stumped by 1. actually LOL Link to comment Share on other sites More sharing options...
GodzillaDeuce Posted April 3, 2012 Share Posted April 3, 2012 no good. sorry :/ Link to comment Share on other sites More sharing options...
rgrewal3 Posted April 3, 2012 Author Share Posted April 3, 2012 no good. sorry :/ Link to comment Share on other sites More sharing options...
GodzillaDeuce Posted April 3, 2012 Share Posted April 3, 2012 i hate the bruins lol but I've been a fan of Recchi since he played for the Blazers Link to comment Share on other sites More sharing options...
rgrewal3 Posted April 3, 2012 Author Share Posted April 3, 2012 i hate the bruins lol but I've been a fan of Recchi since he played for the Blazers Link to comment Share on other sites More sharing options...
GodzillaDeuce Posted April 4, 2012 Share Posted April 4, 2012 well he won a damn cup in another teams uniform too, you gotta change the sig or the avatar Link to comment Share on other sites More sharing options...
Navyblue Posted April 4, 2012 Share Posted April 4, 2012 The beauty of math is that once you know the rules and the language (which don't change), you're set. (not that I know either) Link to comment Share on other sites More sharing options...
rgrewal3 Posted April 4, 2012 Author Share Posted April 4, 2012 The ugliness of math is that grasping the rules and language isn't so easy Link to comment Share on other sites More sharing options...
Guest Posted April 4, 2012 Share Posted April 4, 2012 You're not off the hook in this thread, I'm fully anticipating the answer to number 1 this afternoon! Link to comment Share on other sites More sharing options...
Hobble Posted April 4, 2012 Share Posted April 4, 2012 The fracks I give about math can be found 6x + (pi/4!) meters away. But seriously, it has been a few years since last calculus class :/ Link to comment Share on other sites More sharing options...
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