Posers and Puzzles
26 Feb 08
27 Feb 08
1 edit
Originally posted by eldragonflyNo. Simpsons rule is an approximation for the integeral of a line. Although this may be an application I simply haven't heard of.
i think it's related to finding the area of a circle via an integration, not really sure.
It states that the integeral of a line f(x) between the points a and b is approximatly equal to [(b-a)/6]*[f(a)4*f([a+b]/2)+f(b)]. It has precision 3 (is exact for polynomials of degree 3, e.g. x^3), even though intuitively it is only precise for quadratics.
It is constructed by interpolating the line f(x) through 3 points - f(a), f(b) and the midpoint f([a+b]/2) then integrating this new line. It is not exactly trivial to show that it has precision 3 - it involves integrating the Taylor series and finding the error starts with a term involving x^4.
For the composite form, essentially you split the line using n equally spaced points to get n-1 sub-intervals then apply Simpsons Rule on each sub-interval to get something that's awfully long to type out. It is, however, on wikipedia.
27 Feb 08
Originally posted by eldragonflywhen talking about finding the limit of a series (i'll assume it's in closed form, with n incrementing from some constant to infinity), you will be trying to figure out if the series diverges to infinity or converges to some constant.
can somebody walk me through it please. Also forgot how to take the limit of a series also.
A prerequisite for convergence is that the general term of the series - the general expression in closed form that is being incremented - must have a limit of 0 as n approaches infinity. This is a necessary condition for convergence, but insufficient proof that the series will converge.
Additional tests for seeking proof or disproof of convergence of a series, include but are not limited to: the "power test," the ratio test, the comparison test, the limit comparison test, the improper integral test. Fundamentally, most of these methods basically involve comparing your specific series to more general series whose convergence or divergence is already known. They are attempts at answering the question "does my series act more like a geometric series (convergence), or like a harmonic/arithmetic series (divergence)?" This is my personal take on the idea behind the tests; don't take it as gospel.
if you need specifics on these tests, i could try to elaborate, but i'm sure you'll find more readable resources (.pdf documents, or wikipedia entries) that would be more useful than awkward forum notation 🙂
28 Feb 08
Originally posted by Swlabrthanks.
No. Simpsons rule is an approximation for the integeral of a line. Although this may be an application I simply haven't heard of.
It states that the integeral of a line f(x) between the points a and b is approximatly equal to [(b-a)/6]*[f(a)4*f([a+b]/2)+f(b)]. It has precision 3 (is exact for polynomials of degree 3, e.g. x^3), even though intuitively it i ...[text shortened]... h sub-interval to get something that's awfully long to type out. It is, however, on wikipedia.
28 Feb 08
Originally posted by AetheraelNo that's fine more than sufficient, again thank you.
when talking about finding the limit of a series (i'll assume it's in closed form, with n incrementing from some constant to infinity), you will be trying to figure out if the series diverges to infinity or converges to some constant.
A prerequisite for convergence is that the general term of the series - the general expression in closed form that is be ...[text shortened]... cuments, or wikipedia entries) that would be more useful than awkward forum notation 🙂