Erfan's question at Yahoo! Answers regarding a summation

MarkFL

Staff member
Here is the question:

Summation of series (method of differences_)?

Express the general term in partial fractions and hence find the sum of the series. r/((2r-1)(2r+1)(2r+3)) from 1 to n .
I have posted a link there to this topic so the OP can see my work.

MarkFL

Staff member
Hello Erfan,

We are given to evaluate:

$$\displaystyle S_n=\sum_{r=1}^n\left(\frac{r}{(2r-1)(2r+1)(2r+3)} \right)$$

We are instructed to use partial fraction decomposition on the summand, and so we assume it may be expressed in the form:

$$\displaystyle \frac{r}{(2r-1)(2r+1)(2r+3)}=\frac{A}{2r-1}+\frac{B}{2r+1}+\frac{C}{2r+3}$$

Using the Heaviside cover-up method, we find:

$$\displaystyle A=\frac{\frac{1}{2}}{(2)(4)}=\frac{1}{16}$$

$$\displaystyle B=\frac{-\frac{1}{2}}{(-2)(2)}=\frac{1}{8}$$

$$\displaystyle C=\frac{-\frac{3}{2}}{(-4)(-2)}=-\frac{3}{16}$$

Hence:

$$\displaystyle \frac{r}{(2r-1)(2r+1)(2r+3)}=\frac{1}{16(2r-1)}+\frac{1}{8(2r+1)}-\frac{3}{16(2r+3)}$$

Factoring out $$\displaystyle \frac{1}{16}$$, we may write the sum as follows:

$$\displaystyle S_n=\frac{1}{16}\left(\sum_{r=1}^n\left(\frac{1}{2r-1} \right)+2\sum_{r=1}^n\left(\frac{1}{2r+1} \right)-3\sum_{r=1}^n\left(\frac{1}{2r+3} \right) \right)$$

Re-indexing the sums, we have:

$$\displaystyle S_n=\frac{1}{16}\left(\sum_{r=1}^n\left(\frac{1}{2r-1} \right)+2\sum_{r=2}^{n+1}\left(\frac{1}{2r-1} \right)-3\sum_{r=3}^{n+2}\left(\frac{1}{2r-1} \right) \right)$$

Pulling off terms so that the indices of summation are the same for all three sums, we find:

$$\displaystyle S_n=\frac{1}{16}\left(\left(1+\frac{1}{3}+ \sum_{r=3}^n\left(\frac{1}{2r-1} \right) \right)+\right.$$

$$\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.\left(\frac{2}{3}+2 \sum_{r=3}^n\left(\frac{1}{2r-1} \right)+\frac{2}{2n+1} \right)+\left(-3 \sum_{r=3}^n\left(\frac{1}{2r-1} \right)-\frac{3}{2n+1}-\frac{3}{2n+3} \right) \right)$$

Now the sums all add to zero, and we are left with:

$$\displaystyle S_n=\frac{1}{16}\left(2-\frac{1}{2n+1}-\frac{3}{2n+3} \right)=\frac{n(n+1)}{2(2n+1)(2n+3)}$$