Math Is Fun Forum

3. I dont see any "nice" method, I just multiplied with sin2xcis3x, then used addition formula for sin(2x+x) and cos(2x+x) and then fomrula for double angle. Then using the identity sin^2x+cos^2x=1 to express in only cosx, and then dividing with cosx yields a deg 4 polynomial of cosx. Setting cos^2x=z gives a quadratic polynomial, and I got the final answer for cosx to be:

I think it is one of the so called cauchys functions, which does not need to be proven. Jane, can you state the exact theorems about how differentiation and continuity relates to another? didnt know for example that differentiable at a single point, could be extended to all points, i think i need to repeat this

use the fact:

to bump this thread, here is another sequence:

you dont need induction for the second one, just expand the paranthesis and simplify, and you will end up with an expression that you can easy show that it is divisible by 9 by letting n=3 and n=3k±1. (you can also first change n^3 +(n + 1)^3 + (n+2)^3 to (n - 1)^3 + n^3 + (n+1)^3, which may simplify the expansion)

that was cool jane, but you can also do it really easy with jensen's inequality:
the function

hint: what is the sum of two following terms, ie term n and term n+1?

assume that x+1/x is an integer. Squaring gives:

ps. how do i do the sign, that shows that an element belongs to a set in mathcode?

uhm well yea but i wanted to be pedagogic

aha, I thought he meant:

Hint: Use addition formula (sin(x+y)=sinxcosy+cosxsiny) and then sin2x=2sinxcosx (for cos, cos(x+y)=cosxcosy-sinxsiny, cos2x=cos^2x-sin^2x). And I think you have some errors in the denominator in the second approach.

heres a solution: