FP2.2 - Series

Learning objectives

—FP2.2.1 - Summation of simple finite series n 1 ∑Summation of simple finite series n 1 ∑ using the method of differences.; Students should be able to sum series such as by r(r +1) r=1 1 1 1 using partial fractions such as = −. r(r +1) r r +1.

The method of differences rewrites each term as a difference of shifted expressions. When consecutive terms are displayed, the interior contributions cancel and only a small number of boundary terms remain.

Step	Purpose
Decompose the summand	Use partial fractions, or a supplied identity, to write the term as shifted copies such as $u_r-u_{r+m}$ .
Verify the identity	Recombine the parts before summing so that constants, signs and shifts are correct.
Display boundary terms	Write enough terms from the beginning and end to show exactly what cancels.
Collect survivors	Keep the first $m$ positive terms and the final $m$ negative terms, then simplify.

$\sum_{r=p}^{q}(u_r-u_{r+m})=\sum_{j=0}^{m-1}u_{p+j}-\sum_{j=1}^{m}u_{q+j}$

For the official model, partial fractions give $\frac1{r(r+1)}=\frac1r-\frac1{r+1}.$ Therefore $\sum_{r=1}^{n}\frac1{r(r+1)}=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\cdots+\left(\frac1n-\frac1{n+1}\right).$ Every interior fraction cancels, leaving $1-\frac1{n+1}=\frac{n}{n+1}.$

If a closed form $S(n)=\sum_{r=p}^{n}f(r)$ is already known, a later finite block is found by subtraction: $\sum_{r=a}^{b}f(r)=S(b)-S(a-1).$ This preserves the correct first included term $r=a$ .

Do not claim cancellation without showing the shifted terms, and do not discard every middle-looking term when the shift is $m>1$ : exactly $m$ terms survive at each boundary. Check the starting index before using partial fractions whose denominators could be zero. This objective concerns finite series; an infinite limit requires separate convergence justification and is not asserted here.

Next Topic

—FP2.3 - Further complex numbersRead