Series (Method of Differences) for Edexcel IAL FP2 Further Maths

Edexcel International A-Level Further Mathematics· FP2 2.1 (2018 Spec Issue 3)· 25 min read

1. Core Principle of the Method of Differences★★☆☆☆⏱ 6 min

✓ Calculator OK

📘 Definition

Method of Differences

A summation technique that rewrites each term $u_{r}$ of a series as $u_{r} = f (r) - f (r + k)$ for some function $f$ and positive integer $k$ , so intermediate terms cancel when expanded.

The key advantage of this method is that it eliminates the need to sum every term individually, instead reducing the sum to just a small number of surviving (non-cancelling) terms. The most common case is $k = 1$ , where $u_{r} = f (r) - f (r + 1)$ , leading to a classic telescoping series.

📐 Worked Example

Show that the general term $u_{r} = \frac{1}{r ( r + 1 )}$ can be rewritten as a difference of terms of the form $\frac{1}{r}$ .

1
Use partial fractions to decompose $u_{r}$ :
$\frac{1}{r(r+1)} = \frac{A}{r} + \frac{B}{r+1}$
2
Multiply both sides by $r (r + 1)$ :
$1 = A(r+1) + B r$
3
Substitute $r = 0$ to find $A = 1$ , $r = - 1$ to find $B = - 1$
4
Rearrange to get the difference form:
$\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1} = f(r) - f(r+1) \text{ where } f(r) = \frac{1}{r}$

2. Calculating Finite Series Closed Forms★★★☆☆⏱ 8 min

✓ Calculator OK

Once you have expressed $u_{r}$ as a difference of function values, expand the sum for the first few terms and the last few terms explicitly to identify cancellation patterns. You must show this expansion in your exam answer to get full marks.

Write the sum as $\sum_{r = 1}^{n} [f (r) - f (r + 1)] = \sum_{r = 1}^{n} f (r) - \sum_{r = 1}^{n} f (r + 1)$
Reindex the second sum to start at $r = 2$ : $\sum_{r = 2}^{n + 1} f (r)$
Subtract the two sums: all terms from $r = 2$ to $r = n$ cancel, leaving $f (1) - f (n + 1)$

📐 Worked Example

Find the closed form of $\sum_{r = 1}^{n} \frac{1}{r ( r + 1 )}$ using the method of differences.

1
Use the decomposition we derived earlier:
$\sum_{r=1}^n \frac{1}{r(r+1)} = \sum_{r=1}^n \left(\frac{1}{r} - \frac{1}{r+1}\right)$
2
Expand the first 3 and last 2 terms to show cancellation:
$= \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + ... + \left(\frac{1}{n} - \frac{1}{n+1}\right)$
3
Cancel all intermediate terms: $- \frac{1}{2}$ cancels with $+ \frac{1}{2}$ , $- \frac{1}{3}$ cancels with $+ \frac{1}{3}$ , and so on, up to $- \frac{1}{n}$ cancels with $+ \frac{1}{n}$
4
Write the surviving terms:
$= 1 - \frac{1}{n+1} = \frac{n}{n+1}$

3. Evaluating Sums to Infinity★★★★☆⏱ 7 min

✓ Calculator OK

If the finite sum closed form approaches a finite limit as $n \to \infty$ , the series converges, and this limit is the sum to infinity. For FP2, you must explicitly state which terms vanish as $n \to \infty$ to justify your result.

📘 Definition

Sum to Infinity of a Telescoping Series

The finite limit $L = lim_{n \to \infty} S_{n}$ , where $S_{n}$ is the closed-form finite sum of the first $n$ terms, provided the limit exists.

📐 Worked Example

Find the sum to infinity of $\sum_{r = 1}^{\infty} \frac{1}{r ( r + 1 )}$ , if it exists.

1
Use the finite sum we calculated earlier:
$S_n = \frac{n}{n+1} = 1 - \frac{1}{n+1}$
2
Evaluate $lim_{n \to \infty} S_{n}$ : as $n$ becomes very large, $\frac{1}{n + 1} \to 0$
$\lim_{n \to \infty} S_n = 1 - 0 = 1$
3
Justification: The term $\frac{1}{n + 1}$ vanishes as $n \to \infty$ , so the series converges to 1.

4. Extended Telescoping for Differences with Offset $k>1$★★★★☆⏱ 6 min

✓ Calculator OK

Some questions use a difference with offset $k = 2$ , e.g. $u_{r} = f (r) - f (r + 2)$ . In this case, cancellation leaves 2 terms from the start of the sum and 2 terms from the end, rather than 1 each.

The process remains identical: expand the first $k$ terms and last $k$ terms to identify the surviving terms, then simplify the closed form. For sums to infinity, check that all $n$ -dependent terms vanish as $n \to \infty$ .

📐 Worked Example

Given $u_{r} = \frac{1}{r ( r + 2 )} = \frac{1}{2} (\frac{1}{r} - \frac{1}{r + 2})$ , find $\sum_{r = 1}^{n} u_{r}$ .

1
Write out the expanded sum:
$\sum_{r=1}^n \frac{1}{2}\left(\frac{1}{r} - \frac{1}{r+2}\right) = \frac{1}{2}\left[ \left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + ... + \left(\frac{1}{n-1} - \frac{1}{n+1}\right) + \left(\frac{1}{n} - \frac{1}{n+2}\right) \right]$
2
Identify cancelling terms: $- \frac{1}{3}$ cancels with $+ \frac{1}{3}$ , $- \frac{1}{4}$ cancels with $+ \frac{1}{4}$ , ..., $- \frac{1}{n}$ cancels with $+ \frac{1}{n}$
3
Write surviving terms:
$= \frac{1}{2}\left[ 1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2} \right]$
4
Simplify:
$= \frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$

5. Common Pitfalls

Wrong move:

Only writing the first 2 terms when expanding the series, failing to show cancellation pattern

Why:

Exam markers require explicit evidence you understand telescoping, you will lose 1-2 marks even if your final answer is correct

Correct move:

Write at least the first 3 terms and last 2 terms of the expanded sum to show cancellation clearly

Wrong move:

Confusing the method of differences with FP1 standard sum formulas ( $\sum r$ , $\sum r^{2}$ , etc.) for FP2 series questions

Why:

FP2 series questions explicitly require the method of differences, using standard sums will give the wrong answer and earn zero marks

Correct move:

Always start FP2 series questions by decomposing the general term using partial fractions into a difference of function values

Wrong move:

Forgetting to justify which terms vanish when calculating a sum to infinity

Why:

The exam specification requires you to demonstrate understanding of convergence, missing this justification costs 1 mark

Correct move:

State explicitly that the $n$ -dependent terms tend to 0 as $n \to \infty$ when computing the limit

Wrong move:

Miscalculating the indices of the reindexed sum for offset $k > 1$ , e.g. missing the second surviving term at the start of the sum for $k = 2$

Why:

This leads to an incorrect closed form, often missing a constant term

Correct move:

For offset $k$ , write out the first $k$ and last $k$ terms of the expanded sum to identify all surviving terms

Wrong move:

Using CAS calculator features to compute the sum directly without showing working

Why:

CAS calculators are not permitted in the exam, and you must show all steps of partial fraction decomposition and telescoping to get full marks

Correct move:

Practice working through all steps manually, as you will in the exam

6. Quick Reference Cheatsheet

Scenario	Method Steps	Result Form
Finite sum, $k = 1$ offset	Decompose $u_{r} = f (r) - f (r + 1)$ via partial fractions 2. Expand sum, cancel intermediate terms	$S_{n} = f (1) - f (n + 1)$
Finite sum, $k = 2$ offset	Decompose $u_{r} = f (r) - f (r + 2)$ via partial fractions 2. Expand first 2 and last 2 terms	$S_{n} = f (1) + f (2) - f (n + 1) - f (n + 2)$
Sum to infinity	Compute finite sum $S_{n}$ 2. Evaluate $lim_{n \to \infty} S_{n}$ , justify vanishing terms	$S_{\infty} = lim_{n \to \infty} S_{n}$ , if limit exists

Going deeper

recap_guidePartial Fractions for Edexcel IAL P3Prerequisite skill for decomposing series terms

What's Next

Now that you have mastered the method of differences for FP2 series, you are ready to apply this skill to past paper questions and move on to other core FP2 topics. This technique is often combined with partial fractions in multi-part exam questions, so make sure you are confident decomposing algebraic fractions before attempting timed practice. You can also extend your knowledge to more advanced series topics in university-level mathematics if you are pursuing a STEM degree.