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

> Edexcel International A-Level Further Mathematics · FP2
> Source: https://www.owlsprep.com/study/edexcel-ial-further-math-fp2-series/

This guide teaches the method of differences, the only required series summation technique for Edexcel IAL FP2. You will learn to compute finite closed-form sums and sums to infinity for telescoping series, aligned strictly with the 2018 FP2 specification.

**Prerequisites:** [Partial fraction decomposition (P-level)](https://www.owlsprep.com/study/edexcel-ial-maths-p3-partial-fractions/); [Basic limit evaluation (P-level)](https://www.owlsprep.com/study/edexcel-ial-maths-p1-limits/)

## Learning objectives

- Rewrite series terms as a difference of function values using partial fractions for telescoping
- Sum finite series via the method of differences by identifying and cancelling intermediate terms
- Calculate sums to infinity by evaluating the limit of the finite sum closed form as n→∞, justifying vanishing terms

## Core Principle of the Method of Differences

**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.

> **note**
>
> For FP2 series questions, you will always first decompose $u_r$ into the required difference form using partial fractions, a prerequisite skill from P3 Mathematics.

**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:

   $$\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1} = f(r) - f(r+1) \text{ where } f(r) = \frac{1}{r}$$

*Calculator:* allowed

## Calculating Finite Series Closed Forms

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.

1. Write the sum as $\sum_{r=1}^n [f(r) - f(r+1)] = \sum_{r=1}^n f(r) - \sum_{r=1}^n f(r+1)$
2. Reindex the second sum to start at $r=2$: $\sum_{r=2}^{n+1} f(r)$
3. 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:

   $$= 1 - \frac{1}{n+1} = \frac{n}{n+1}$$

> **Exam Tip**
>
> Always write out at least the first 3 terms and last 2 terms of the expanded series in your exam answer to demonstrate you understand the cancellation, even if you can compute the result quickly.

*Calculator:* allowed

## Evaluating Sums to Infinity

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.

**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.

> **warning**
>
> A sum to infinity only exists if the terms of $S_n$ that depend on $n$ tend to 0 as $n \to \infty$. If they do not, the series diverges and has no sum to infinity.

**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.

*Calculator:* allowed

## Extended Telescoping for Differences with Offset $k>1$

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 $u_r = \frac{1}{r(r+2)} = \frac{1}{2}\left(\frac{1}{r} - \frac{1}{r+2}\right)$, 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:

   $$= \frac{3}{4} - \frac{2n+3}{2(n+1)(n+2)}$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Only writing the first 2 terms when expanding the series, failing to show cancellation pattern
  - Why it fails: Exam markers require explicit evidence you understand telescoping, you will lose 1-2 marks even if your final answer is correct
  - Correct: Write at least the first 3 terms and last 2 terms of the expanded sum to show cancellation clearly
- **Wrong:** Confusing the method of differences with FP1 standard sum formulas ($\sum r$, $\sum r^2$, etc.) for FP2 series questions
  - Why it fails: FP2 series questions explicitly require the method of differences, using standard sums will give the wrong answer and earn zero marks
  - Correct: Always start FP2 series questions by decomposing the general term using partial fractions into a difference of function values
- **Wrong:** Forgetting to justify which terms vanish when calculating a sum to infinity
  - Why it fails: The exam specification requires you to demonstrate understanding of convergence, missing this justification costs 1 mark
  - Correct: State explicitly that the $n$-dependent terms tend to 0 as $n \to \infty$ when computing the limit
- **Wrong:** 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 it fails: This leads to an incorrect closed form, often missing a constant term
  - Correct: For offset $k$, write out the first $k$ and last $k$ terms of the expanded sum to identify all surviving terms
- **Wrong:** Using CAS calculator features to compute the sum directly without showing working
  - Why it fails: 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: Practice working through all steps manually, as you will in the exam

## Cheatsheet

| Scenario | Method Steps | Result Form |
| --- | --- | --- |
| Finite sum, $k=1$ offset | 1. 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 | 1. 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 | 1. 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 |

## 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.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/edexcel-ial-further-math-fp2-series/