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I have a question regarding a proof by induction. We have to see whether or not the following series converges. $$U_n = \frac{1 \cdot 4 \cdot 7 \cdots (3n - 2)}{2 \cdot 5 \cdot 8 \cdots (3n-1)}$$ I was trying to do this by proving that this series has a lower limit of $0$ and is decreasing. It's easy enough to see that it has a lower limit of $0$ but proving (by induction) that this series is decreasing has proven to be difficult. I understand that I need to prove that $u_n > u_{n+1}$ but I have no idea how to go about doing this.

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  • $\begingroup$ It is not a series. $\sum_{n\geq 1}a_n$ is a series. $\endgroup$ – Jack D'Aurizio May 22 '16 at 15:33
  • $\begingroup$ Is this the $n^{th}$ term of a series? $\endgroup$ – Qwerty May 22 '16 at 15:45
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Let $a_n = \dfrac{1*4* .. *(3n - 2)}{2*5 .. *(3n - 1)}$.

The sequence converges from the Monotonic Convergence Theorem, as it is bounded below (by zero) and is decreasing:

$a_{n + 1} = a_n\dfrac{3n + 1}{3n + 2} < a_n$

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$$u_n>\frac{3n-2}{3n-1}u_n=u_{n+1}$$ and so by induction the entire sequence is monotonic decreasing.

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