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I need to prove that the following converges: $$\sum\limits_{n=0}^{\infty} n^3e^{-n^4}$$

I'm not sure which test to use. I've tried ratio test but it gets messy. I don't think I have any sums to use in the comparison test and the integral test obviously wouldn't work. Thanks

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This series is quite easy: simply apply ANY convergence test. – Crostul Mar 25 at 14:03
    
I got rid of a stray equals sign from the summand. Is the currently displayed sum what you intended? – Patrick Stevens Mar 25 at 14:05
    
@PatrickStevens Yes it is, thanks – Alex Mar 25 at 14:06

Ratio test works fine. The ratio in question is $$\frac{(n+1)^3 e^{-(n+1)^4}}{n^3 e^{-n^4}} = \left( \frac{n+1}{n} \right)^3 e^{n^4-(n+1)^4} = \left( \frac{n+1}{n} \right)^3 e^{-1-4n -6n^2 -4n^3} $$ The exponential term tends to $0$ as $n \to \infty$; the fraction tends to $1$; so the ratio of the terms tends to $0$.


Integral test works fine: $$\int_1^{\infty} x^3 e^{-x^4} dx = \int_1^{\infty}\frac{1}{4} \dfrac{d}{dx}(e^{-x^4}) dx = \frac{1}{4e}$$ To verify that the integrand is decreasing for sufficiently large $x$, differentiate it, obtaining $$3x^2 e^{-x^4} - 4 x^6 e^{-x^4} = x^2 e^{-x^4}(3-4x^4)$$ clearly negative for $x > 1$.


Comparison test works fine: since $e^{-n^4} \leq \frac{1}{n^5}$, have $$n^3 e^{-n^4} \leq n^{-2}$$ It is true that $e^{-n^4} \leq \frac{1}{n^5}$, because that's equivalent to $$n^5 e^{-n^4} \leq 1$$ Differentiating the left-hand side, obtain $$n^4(e^{-n^4}) (5-4n^4)$$ which is negative for integer $n > 1$; so the maximum of the function $n^5 e^{-n^4}$ over the integers is attained at $n=1$, and then we do indeed get $e^{-1} \leq 1$.

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oops idk what I was thinking lol. Thanks! Edit: Doesn't it have to be all decreasing to apply the integral test? – Alex Mar 25 at 14:06
    
@Alex Yes. Easily checked; I've put that in. – Patrick Stevens Mar 25 at 14:14
    
I would appreciate it if the downvoter would tell me what I've got wrong, because I can't see the mistake. – Patrick Stevens Mar 25 at 14:19
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Solid answer(s) ... +1 Note without much effort, $e^{x}\ge 1+x$ gives $e^{-n^4}\le \frac{1}{n^4+1}<\frac{1}{2n^4}$ for $n\ge 1$. ;-)) -Mark – Dr. MV Mar 25 at 14:40

Directly $\;n-$th root test:

$$\sqrt[n]{n^3e^{-n^4}}=\frac{\left(\sqrt[n]n\right)^3}{e^{n^3}}\xrightarrow[n\to\infty]{}0<1$$

and thus the series converges

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First note that $$\sum\limits_{n=0}^{\infty} \left|\frac{n^3}{e^{n^4}}\right|=\sum\limits_{n=1}^{\infty} \left|\frac{n^3}{e^{n^4}}\right|\leq\sum\limits_{n=1}^{\infty} \left|\frac{n^3}{n^5}\right|=\sum\limits_{n=1}^{\infty} \left|\frac{1}{n^2}\right|$$ Therefore $$\sum\limits_{n=0}^{\infty} \frac{n^3}{e^{n^4}}=\mbox{absolutely convergent}$$

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The form of the term seems to ask for the Integral Test since $$ \int_x^\infty t^3e^{-t^4}\,\mathrm{d}t=\tfrac14e^{-x^4} $$

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