Erfan's question at Yahoo Answers regarding a summation

[MarkFL]

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]

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:

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