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$ Find whether each of the following series converge or diverge $ $ a) $ \sum_{k = 1}^{\infty}{\sin^2{\frac{1}{k}}} $ b) $ \sum_{k = 1}^{\infty}{\frac{\ln{k!}}{k^3}} $ c) $ \sum_{k = 1}^{\infty}{\frac{(\ln{k})^3}{}{\sqrt{k^3-3k^2 +1}}}

From my knowledge 1 is completely wrong. If a_k diverges then that can prove that b_k diverges. Also from my knowledge of 2 if the ratio is 1 then we have to do integral test? EDIT: and for 3 we just have to prove a_n > a_{n+1}, a_{n} -> 0 as n -> infty, a_n >= 0?

Okay... I have questions about convergence and divergence. 1) Is this a sufficient condition to prove convergence of b_k: \sum_{k=0}^{\infty}a_k < \sum_{k=0}^{\infty}b_k $ If we know $ \sum_{k=0}^{\infty}a_k $ converges $ 2) To prove that \ \sum_{k=0}^{\infty}a_k converges we...

If we are given the eigenvalues of the matrix with corresponding eigenvectors what is the simplest way to find the original matrix without doing A = MDM^(-1)?

$ If $ u(x,t) = f(x+tu(x,t)) $ prove that $ u(x,t) $ is a solution to $ \frac{du}{dt} = u \frac{du}{dx} EDIT: nevermind got it

Thanks guys :) ! Will be back for more soon, I pretty much skipped 70-80 % of my lectures so I need your help !

$ 1) T is a linear transformation in $ L(U,V) $ and $ u_1, u_2,..u_n \in U. $ Given that $ T(u_1), T(u_2)...T(u_n) $ are linear independent elements of V, prove that $u_1,u_2...u_n $ are also linear independent. Also prove the converse is false. $ $ 2) Prove that the Eigenvalues of an n x n...

my bad...

What's f(x) ?

Re: HSC 4U Integration Marathon 2017 \int{\frac{x^2 -1}{x^2 + 1} \cdot \frac{1}{\sqrt{1+x^4}}}dx

Re: HSC 4U Integration Marathon 2017 Learn Latex pls

a_n = a_{n-1} + a_{n-2} + ... + a_1 + a_0

The last question loops like one of the BOS trial questions...

I think it will be a summation questions to prove a taylor series or something.

Re: HSC 4U Integration Marathon 2017 $ Let $ x = \sin^2{\theta} $ and then do IBP to get $ I = \frac{\pi}{4} Chose otherwise soz. I can see the borders way but why bother...

Re: HSC 4U Integration Marathon 2017 If the integrals easy enough to inspect such as a reverse product rule then you can... You can't just do what Paradoxica does and just say for question 16 integral "by inspection...".

Re: MX2 2016 Integration Marathon Offfffttt relaxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx....

Re: HSC 2017 4U Marathon Yes.

Re: HSC 2017 4U Marathon Ill get the ball rolling... THE SOCCERBALL ROLLING!!! YOU GET IT?????! pls save me from finals... i) $ Let $ z = x + iy $ and $ w = a + ib $ By inspection notice that: $ (ay - bx)^2 > 0 a^2y^2 -2abxy +b^2x^2 > 0 a^2x^2 +a^2y^2 +b^2x^2 +b^2y^2 > a^2x^2...

Re: HSC 2017 4U Marathon \frac{z + 1}{z-1} \cdot \frac{\bar{z} -1}{\bar{z} - 1} \frac{z\bar{z} - z + \bar{z} - 1}{z\bar{z} -z - \bar{z} + 1} $ Take note that : $ z\bar{z} = x^2 + y^2 \frac{x^2 + y^2 - 1 - 2iy }{x^2 + y^2 -2x + 1} \frac{x^2 + y^2 -1}{x^2 + y^2 +1 -2x} -2i \cdot...