Sum the series below:
[MATH]\sum_{n=4}^{999}\dfrac{1}{\sqrt[3]{n^2-2n+1}+\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}}.[/MATH]
Question 1: Do you think the expression on the denominator can be factored?
Yes.
No.
This is a really rich way to asking problem so we make the students to think deep before simply giving out the answer yes or no so easily.
Showing posts with label denominator. Show all posts
Showing posts with label denominator. Show all posts
Putnam Math Exam Problem: Evaluate [MATH]\prod_{n=2}^{\infty}\frac{n^3-1}{n^3+1}[/MATH]
Putnam Math Exam Problem:
Evaluate [MATH]\prod_{n=2}^{\infty}\frac{n^3-1}{n^3+1}[/MATH].
I admit it, the first (or should I say, the only) thing that we could do about the given fraction's expression, is to factor it:
Evaluate [MATH]\prod_{n=2}^{\infty}\frac{n^3-1}{n^3+1}[/MATH].
I admit it, the first (or should I say, the only) thing that we could do about the given fraction's expression, is to factor it:
Simplify $\dfrac{\cos 1^{\circ}+\cos 2^{\circ}+\cdots+\cos 44^{\circ}}{\sin 1^{\circ}+\sin 2^{\circ}+\cdots+\sin 44^{\circ}}$
Express $\dfrac{\cos 1^{\circ}+\cos 2^{\circ}+\cdots+\cos 44^{\circ}}{\sin 1^{\circ}+\sin 2^{\circ}+\cdots+\sin 44^{\circ}}$ in the form $a+b\sqrt{c}$, where $a,\,b,\,c$ are positive integers.
(This question showed up here(math-teacher-guide) but I did not include its solution in that slide show).
My solution:
Normally for simplifying trigonometric question such as this one, there is one thing that's so worth noticing:
(This question showed up here(math-teacher-guide) but I did not include its solution in that slide show).
My solution:
Normally for simplifying trigonometric question such as this one, there is one thing that's so worth noticing:
Olympiad Trigonometry Problem
Prove $\tan^2 20^\circ+\tan^2 40^\circ+\tan^2 80^\circ=33$.
Good trigonometry problem is hard to come by, and when we, the math educator found one, we have to take advantage of it and make full use of it.
Of course, we can rest assured that this trigonometry problem can be tackled using the sum-to-product and product-to-sum identities in a really messy and tedious way. After all, any given math problem can be solved in the most traditional way, isn't it?
Good trigonometry problem is hard to come by, and when we, the math educator found one, we have to take advantage of it and make full use of it.
Of course, we can rest assured that this trigonometry problem can be tackled using the sum-to-product and product-to-sum identities in a really messy and tedious way. After all, any given math problem can be solved in the most traditional way, isn't it?
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