Two problems that I never learned how to solve

First one

[1 - 2] + [ 3 - 4] + ... + [2005 - 2006] + 2007

Note  that  each  pair of terms in the brackets  sum to  -1

And we have   2006/2  =  1003 of these pairs....so the series sums to  :

1003 (-1)  +  2007    =  1004  

Second one.....

2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5) =

2/6   +  2/24   +  2/60  =

1/3   +  1/12  +  1/30  =        

20/60   +  5/60  +  2/60   =

27  / 60 

Sorry, ACG...I know that the infinite sum of the second series is  1/2, but I don't know how to prove it......heureka  may know how to prove this.....I'll  send him a message....

∑2/[n(n+1)(n+2)] from n=1 to infinity, then it converges to 1/2.

Sorry, same here, except that the Partial Sum Formula would be:  (n^2 + 3n)/(2 (n + 1) (n + 2)). So, if n=1,000,000, then:[1,003,000/2,006,004] =0.49999900.............etc.

Compute the sum    2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5)+...

see link: http://web2.0calc.com/questions/algebra_47009

In General:

\(\begin{array}{lcll} s = \dfrac{1}{1 \cdot 2 } + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4 } + \dfrac{1}{4 \cdot 5 } + \cdots \ + \dfrac{1}{n \cdot (n+1)} + \cdots =\ \frac{1}{1!0!}\cdot \frac{1}{1} = 1 \\\\ s =\mathbf{ \dfrac{1}{1 \cdot 2 \cdot 3} + \dfrac{1}{2 \cdot 3 \cdot 4} + \dfrac{1}{3 \cdot 4 \cdot 5} + \dfrac{1}{4 \cdot 5 \cdot 6} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2)} + \cdots =\ \frac{1}{2!0!}\cdot \frac{1}{2} } = \frac{1}{4} \\\\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6} + \dfrac{1}{4 \cdot 5 \cdot 6 \cdot 7} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2) \cdot (n+3)} + \cdots =\ \frac{1}{3!0!}\cdot \frac{1}{3} = \frac{1}{18} \\\\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \dfrac{1}{4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2) \cdot (n+3 \cdot (n+4) } + \cdots =\ \frac{1}{4!0!}\cdot \frac{1}{4} = \frac{1}{96} \\\\ \ldots \\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdots } + \dfrac{1}{2 \cdot 3 \cdot 4 \cdots } + \dfrac{1}{3 \cdot 4 \cdot 5 \cdots} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2)\cdot~\cdots ~\cdot (n+m) } + \cdots =\ \frac{1}{m!0!}\cdot \frac{1}{m} = \frac{1}{m\cdot m!} = \frac{1}{(m+1)!-m!} \\\\ \end{array} \\ \)