Summation of Series (Further Pure Mathematics 1, Topic 1.3)
Welcome to the fascinating world of Summation of Series! In Further Mathematics, we move beyond simple arithmetic and geometric progressions to tackle much more complex sums. This chapter gives you the powerful tools needed to find the exact sum of a series, often involving products of terms, not just simple additions.
Don't worry if this feels intimidating. We will break down these methods into two key areas: using standard building blocks (polynomial sums) and using a clever trick called the Method of Differences (telescoping series). Let’s get counting!
1. The Standard Results: Your Building Blocks
The Further Maths syllabus requires you to know and use the summation formulas for powers of \(r\). These results are typically provided in the MF19 Formula Booklet, but knowing them well saves time and helps you understand related problems. They are the algebraic 'building blocks' for solving complex summation problems.
The sum of the first \(n\) terms of a series is often denoted by \(S_n = \sum_{r=1}^{n} U_r\). For the standard results, \(U_r = r^k\):
A. Sum of the First \(n\) Integers: \(\sum r\)
This is simply the sum of an Arithmetic Progression (AP).
\[ \sum_{r=1}^{n} r = 1 + 2 + 3 + \dots + n = \frac{1}{2}n(n+1) \]
B. Sum of the First \(n\) Squares: \(\sum r^2\)
\[ \sum_{r=1}^{n} r^2 = 1^2 + 2^2 + \dots + n^2 = \frac{1}{6}n(n+1)(2n+1) \]
C. Sum of the First \(n\) Cubes: \(\sum r^3\)
\[ \sum_{r=1}^{n} r^3 = 1^3 + 2^3 + \dots + n^3 = \frac{1}{4}n^2(n+1)^2 \]
💡 Memory Aid: Notice the neat relationship between \(\sum r\) and \(\sum r^3\).
\[ \sum_{r=1}^{n} r^3 = \left( \sum_{r=1}^{n} r \right)^2 \]
Quick Review: Standard Sums
- \(\sum r = \frac{1}{2}n(n+1)\)
- \(\sum r^2 = \frac{1}{6}n(n+1)(2n+1)\)
- \(\sum r^3 = (\sum r)^2\)
2. Finding Related Sums (Linearity of Summation)
Often, you won't be asked to find the sum of just \(r^2\). Instead, the general term \(U_r\) will be a polynomial in \(r\), such as \(U_r = r(r+1)\) or \(U_r = 3r^2 - 5r + 2\).
The Principle of Linearity
The most important tool here is the principle of Linearity. It states that summation can be applied term-by-term, and constants can be factored out. If \(U_r\) is a combination of other terms, you can split the sum up:
\[ \sum (A U_r + B V_r) = A \sum U_r + B \sum V_r \]
Step-by-Step Process for Related Sums:
- Expand \(U_r\): Rewrite the general term \(U_r\) as a polynomial in \(r\).
Example: If \(U_r = r(r+1)(r-1)\), expand it to \(U_r = r^3 - r\). - Split the Sum: Use linearity to separate the sum into terms involving \(\sum r^3\), \(\sum r^2\), \(\sum r\), and constant terms.
Example: \(\sum_{r=1}^{n} (r^3 - r) = \sum_{r=1}^{n} r^3 - \sum_{r=1}^{n} r\) - Apply Standard Results: Substitute the known formulas for \(\sum r\), \(\sum r^2\), etc.
- Simplify: Factorize the resulting expression to get the simplest possible final answer for \(S_n\).
Important Note: Always ensure the summation starts at \(r=1\). If the sum starts at \(r=k\) (where \(k>1\)), you must calculate \(\sum_{r=1}^{n} U_r - \sum_{r=1}^{k-1} U_r\).
3. The Method of Differences (Telescoping Series)
What if the series is not a simple polynomial in \(r\)? For example, sums involving fractions or trigonometric functions often require the Method of Differences (also known as the telescoping method). This is a crucial skill for Further Maths.
The Analogy: The Telescoping Stick
Imagine an old-fashioned telescope or a folding walking stick. When you open it, many parts are visible. But when you collapse it, only the first and last parts remain visible. The Method of Differences works by expressing the general term \(U_r\) as a difference between two consecutive functions, $f(r)$ and $f(r+1)$, such that most terms cancel each other out when summed.
We aim to write \(U_r\) in the form: \[ U_r = f(r) - f(r+k) \text{ (where } k \text{ is a small integer)} \]
Prerequisite Skill: Partial Fractions
For series involving fractions (like \(\frac{1}{r(r+1)}\)), the first step is almost always decomposing the term using partial fractions to achieve the required difference form.
Example: Let \(U_r = \frac{1}{r(r+1)}\). We use partial fractions: \[ U_r = \frac{1}{r} - \frac{1}{r+1} \]
Here, $f(r) = \frac{1}{r}$ and $f(r+1) = \frac{1}{r+1}$. This is the perfect difference form!
Step-by-Step Summation using Differences:
Assume we are summing \(U_r = f(r) - f(r+1)\) from \(r=1\) to \(n\).
- Express \(U_r\) as a Difference: (Use partial fractions if necessary). Ensure \(U_r\) is in the form \(f(r) - f(r+k)\).
- List the Terms: Write out the first few terms and the last few terms of the series \(S_n\).
\[ S_n = U_1 + U_2 + U_3 + \dots + U_{n-1} + U_n \]
Substituting \(U_r = f(r) - f(r+1)\):
\[ U_1 = f(1) - f(2) \]
\[ U_2 = f(2) - f(3) \]
\[ U_3 = f(3) - f(4) \]
\[ \dots \]
\[ U_{n-1} = f(n-1) - f(n) \]
\[ U_n = f(n) - f(n+1) \]
You can see the diagonal cancellation: the \(-f(2)\) cancels with the \(+f(2)\), \(-f(3)\) cancels with \(+f(3)\), and so on.
- Identify Remaining Terms: After cancellation (the telescoping), only the terms at the beginning and the end remain.
\[ S_n = f(1) - f(n+1) \]
Key Takeaway for Method of Differences: The sum $S_n$ is the difference between the first few terms of \(f(r)\) and the last few terms of \(f(r+k)\).
4. Sums to Infinity and Convergence
Not all series have a finite sum. If the terms keep getting bigger, the sum will tend to infinity (it diverges). We are interested in series that converge—meaning they settle down to a finite, fixed value when we sum an infinite number of terms.
A. Definition of Convergence
A series \(\sum_{r=1}^{\infty} U_r\) is convergent if the sum of its first \(n\) terms, \(S_n\), tends towards a finite limit \(L\) as \(n \to \infty\).
\[ S_{\infty} = \lim_{n \to \infty} S_n \]
If this limit exists and is finite, the series converges, and $L$ is the Sum to Infinity (\(S_{\infty}\)).
B. Finding \(S_{\infty}\) for Telescoping Series
For series solved using the Method of Differences, finding \(S_{\infty}\) is straightforward once you have \(S_n\).
- Find \(S_n\): Use the Method of Differences to determine the algebraic expression for \(S_n\).
Example: \(S_n = 1 - \frac{1}{n+1}\) - Take the Limit: Find the limit of \(S_n\) as \(n\) approaches infinity.
Continuing the example: \[ S_{\infty} = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) \]
As \(n \to \infty\), the term \(\frac{1}{n+1}\) tends to \(0\).
\[ S_{\infty} = 1 - 0 = 1 \]
Since \(S_{\infty}\) is a finite value (1), the series converges.
Common Trap: If one of the remaining terms in \(S_n\) (like \(f(n)\)) contains \(n\) in the numerator, that term will usually tend to infinity as \(n \to \infty\), and the series will diverge. Always carefully check the limit of all non-cancelling terms.
Did You Know?
The Method of Differences is sometimes used in unexpected contexts, such as summing specific trigonometric series. In Further Maths, you might encounter terms like \(U_r = \tan(r) - \tan(r-1)\) where the terms also collapse like a telescope!
Key Takeaway: Convergence for these series depends entirely on whether the limits of the remaining end terms of \(S_n\) are finite as \(n \to \infty\). If they are, the series converges to that limit.
Chapter Summary
You have mastered two fundamental techniques for summing series in Further Mathematics:
1. Polynomial Sums: Use the Standard Results (\(\sum r, \sum r^2, \sum r^3\)) and the principle of linearity (splitting the sum and factoring out constants) to find \(S_n\).
2. Telescoping Sums: Use the Method of Differences. This requires expressing the general term \(U_r\) as \(f(r) - f(r+k)\), often using partial fractions. When summed, terms cancel out, leaving only a few beginning and ending terms to form \(S_n\).
3. Sum to Infinity: For convergent series, find \(S_{\infty}\) by taking the limit of \(S_n\) as \(n \to \infty\).
Keep practising your algebraic simplification skills—they are essential when using the standard results—and your partial fraction decomposition—it’s crucial for the Method of Differences! You've got this!