Quod Erat Demonstrandum

2015/08/28

sum of 1/k^2 from 1 to infinity

Filed under: Additional / Applied Mathematics,Fun — johnmayhk @ 4:30 下午
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非正式地所謂證明 \displaystyle \sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}
…………………………………………….

首先要知,下式的根之總和(sum of roots)

ax^n+bx^{n-1}+\dots +cx+d=0 ………. (*)

-\frac{b}{a}(more…)

廣告

2008/10/10

Say something about series in Applied Mathematics (II)

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 4:53 下午
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Here is just a typical question in AL Applied Mathematics (II).

For natural numbers m, n (m \ge n).

Let f(x) = x^ne^x, evaluate f^{(m)}(0). (more…)

2008/09/29

Just a question of applied math. from a F.7 boy

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 6:13 下午
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Just discuss an easy AL Applied Mathematics (II) Question with students.

Let f(x), g(x), h(x) be twice differentiable functions such that f(x) = g^2(x) + x^3h(x).

(a) Let p(x) = \frac{f(x)}{g(x)}, where g(0) \ne 0. Show that p(0) = g(0), p'(0) = g'(0), p''(0) = g''(0).

(b) Using (a), or otherwise, find Taylor’s expansion of the function \frac{2x^4 - 3x^3 + x + 4}{\sqrt{x + 4}} about x = 0, up to the term in x^2. (more…)

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