How do I write these fractions as a series (if possible)?

Question

idku · Answer

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

idku · Answer

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

idku · Answer

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....

idku · Answer

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

anonymous · Answer

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

idku · Answer

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 !!!!

anonymous · Answer

yw

idku · Answer

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

idku · Answer

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