6
$\begingroup$

Please suggest an approach for this task.

  • 2
    Type "sum 1/(n(n+1)(n+2)(n+3))" into wolfram alpha.2010-09-27

3 Answers 3

9

$\dfrac{1}{k(k+1)(k+2)(k+3)} = \dfrac{1}{3} (\dfrac{k+3}{k(k+1)(k+2)(k+3)} - \dfrac{k}{k(k+1)(k+2)(k+3)})$ $ = \dfrac{1}{3}(\dfrac{1}{k(k+1)(k+2)} - \dfrac{1}{(k+1)(k+2)(k+3)})$

$\sum_{k=1}^{\infty}\dfrac{1}{k(k+1)(k+2)(k+3)} = \dfrac{1}{3} \dfrac{1}{2*3} = \dfrac{1}{18}$

@moron Yes, you are right. For the first n terms:

$\sum_{k=1}^{n}\dfrac{1}{k(k+1)(k+2)(k+3)} = \dfrac{1}{3} (\dfrac{1}{1*2*3} - \dfrac{1}{(n+1)(n+2)(n+3)})$

[edit] more details

$\sum_{k=1}^{n}\dfrac{1}{k(k+1)(k+2)(k+3)} = \sum_{k=1}^{n} \dfrac{1}{3} (\dfrac{1}{k(k+1)(k+2)} - \dfrac{1}{(k+1)(k+2)(k+3)})$ $= \dfrac{1}{3} [\sum_{k=1}^{n} \dfrac{1}{k(k+1)(k+2)} - \sum_{k=1}^{n} \dfrac{1}{(k+1)(k+2)(k+3)}]$ $= \dfrac{1}{3} [\sum_{k=0}^{n-1} \dfrac{1}{(k+1)(k+2)(k+3)} - \sum_{k=1}^{n} \dfrac{1}{(k+1)(k+2)(k+3)}]$ $= \dfrac{1}{3} (\dfrac{1}{1*2*3} - \dfrac{1}{(n+1)(n+2)(n+3)})$

  • 2
    +1: Even though the question says first n terms, this can easily be adapted for that.2010-09-27
  • 0
    @Moron Thank for your comment2010-09-27
  • 1
    @Branimir: You are welcome! And I see you joined today, welcome to this site :-)2010-09-27
  • 0
    so tricky!!! :)2010-09-27
  • 0
    Say n = 2, then 1/24 + 1/180 = 1/20 where as your solution is giving 1/18 - 1/72 = 1/24 ?!2010-09-27
  • 0
    Fixed typo in last expression. Is it ok now?2010-09-27
  • 0
    t's working but unfortunately my current skills is not sufficient to understand your solution (at-least the form it is answered) :) I will be grateful to you if you could elaborate a bit more.2010-09-27
  • 0
    You challenged my tex skills :) Try to expand expression for example for n=3, you will see that second addend for "k" is the same but with opposite sign as first addend for "k+1".2010-09-27
8

This is similar to what has been said by Branimir, but shows how we can extend the result to $$\sum_{k=1}^n {1 \over k(k+1) \cdots (k+m)}, \qquad m \in \mathbb{N}.$$

We can build up the result from the identities $${1 \over k(k+1)} = {1 \over k} - { 1 \over k+1}, \qquad (1)$$ $${1 \over k(k+1)(k+2)} = {1 \over 2} \left( {1 \over k(k+1)} - { 1 \over (k+1)(k+2)} \right),$$ $${1 \over k(k+1)(k+2)(k+3)} = {1 \over 3} \left( {1 \over k(k+1)(k+2)} - { 1 \over (k+1)(k+2)(k+3)} \right), \quad \textrm{ etc...}$$

Write $S_1 = \sum_{k=1}^n {1 \over k(k+1)},$ $S_2 = \sum_{k=1}^n {1 \over k(k+1)(k+2)},$ etc

Summing for $S_1$ using (1) all terms on RHS cancel to get the classic $$S_1 = \sum_{k=1}^n {1 \over k(k+1)} = 1 – {1 \over n+1} = {n \over n+1}.$$ We then sum the series for $S_2$ using this result obtained for $S_1,$ and so on.

  • 2
    This can be expressed by saying that "negative falling powers behave nicely with respect to the difference operator". See p.53 in "Concrete Mathematics" by Graham, Knuth & Patashnik.2010-09-27
5

HINT $\rm\displaystyle\ \frac{1}{(k+1)(k+2)(k+3)(k+4)} = \frac{1}{6(k+1)} - \frac{1}{2(k+2)}+\frac{1}{2(k+3)}-\frac{1}{6(k+4)}$

$\rm\ f(k+1)-f(k)\: = $ above $\rm\displaystyle\ \Rightarrow\ f(k) \:=\: c_0 + \frac{c_1}{k+1}\ \:+\:\ \frac{c_2}{k+2}\ \:+\:\ \frac{c_3}{k+3}$

Calculating yields $\rm\ c_0,c_1,c_2,c_3 \ =\ 1/18,\ -1/6,\ 1/3,\ -1/6$.

For remarks on the group theory behind rational indefinite summation see my post here