<s>[INST] Sum of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $ What is the limit of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $?
The $n$th summand is $\frac{1}{(2n)(2n + 1)(2n+2)} = \frac{1}{4} \frac{1}{n(2n+1)(n+1)}$.
I have tried expressing this as a telescoping sum, or as the limit of Riemann sums of a partition (the usual methods I normally try when doing this type of question- what are some other strategies?)
 [/INST] You could try the generating form
$$F(x)=\frac{x^4}{2.3.4}+\frac{x^6}{4.5.6}+...\\
\frac{d^3F}{dx^3}=x+x^3+x^5+...=\frac x{1-x^2}$$
Try to integrate the last expression three times, then take the limit as $x\to1$
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