## idku one year ago How do I write these fractions as a series (if possible)?

1. idku

$$\large\color{black}{ \displaystyle \frac{ 1 }{2\times 3 \times 4} -\frac{ 1 }{4\times 5 \times 6}+\frac{ 1 }{6\times 7 \times 8}-\frac{ 1 }{8\times 9 \times 10}~+.... }$$ and this pattern continues like this.

2. idku

$$\Large\color{black}{ \displaystyle \sum_{n=1}^\infty ~\left[\frac{\left(½-(½)(-1)^n\right)}{(n+1)(n+2)(n+3)}~\right] }$$ but there is one problem here

3. idku

I need to have: positive output, when n=1 negative output, when n=3 positive output, when n=5 negative out, when n=7 so on....

4. idku

I need alternation, but regular (-1)$$^n$$ wouldn't suffice here....

5. anonymous

$\sum_{n=1}^{\infty}\frac{ \left( -1 \right)^{n+1} }{ 2n \left( 2n+1 \right)\left( 2n+2 \right) }$

6. idku

oh yeah! I thought of 2n in the beginning, but for some reason I thought it was wrong... clearly, 2*3*4 then 4*5*6 and on also, the alternation is there. THANKS !!!!

7. anonymous

yw

8. idku

So, I can say: $$\large \displaystyle \pi =3+\sum_{n=1}^{\infty}\frac{ 4\left( -1 \right)^{n+1} }{ 2n \left( 2n+1 \right)\left( 2n+2 \right) }$$

9. idku

this is why i wanted that series representation. thanks once again:)