Find out where this power series converges

tamtam402
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Homework Statement


Find out where this power series converges.

Ʃ(xn2n) / (3n + n3)


Homework Equations





The Attempt at a Solution



I'm trying to use the ratio test to solve it. I end up with the following equation, which I am unable to reduce further:

pn = 2x (3n + n3)/[(3)(3)n+n3(1+1/n)3]
 
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My guess is, since 3^n goes to infinity faster than n^3 (exponentials are faster than polynomials), is that your ratios go to \frac{2}{3}x. Tnen you want |x|<\frac{3}{2}. Not sure what happens at the boundaries. To check the limit I guessed at, maybe use l'Hopital's rule 3 times?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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