Series

Welcome to the World of Series!

In this chapter of Core Pure, we are going to explore the art of adding things up—sometimes a few things, and sometimes infinitely many! You’ve already met sequences and series in your standard A Level Maths, but here in Further Maths, we’re going to give you some "superpowers" to handle much more complex sums. Whether it’s finding a shortcut for adding the first million square numbers or approximating a complicated curve with a simple polynomial, you're about to learn some of the most beautiful and useful tools in mathematics.

1. The Basics: Sequences vs. Series

Before we dive in, let’s make sure we’re on the same page.
A sequence is just a list of numbers in a specific order (like 2, 4, 6, 8...).
A series is what you get when you add those numbers together (like \(2 + 4 + 6 + 8\)).

Quick Review:
- Converge: A series converges if the sum approaches a specific, finite number as you add more and more terms.
- Diverge: If the sum keeps growing forever (to infinity), we say it diverges.

2. Standard Formulae: The "Shortcuts"

Sometimes, adding numbers one by one is too slow. Imagine your teacher asks you to add all the square numbers from \(1^2\) up to \(100^2\). You could be there all day! Instead, we use standard formulae.

For the sum of the first \(n\) terms, we use the Greek letter \(\Sigma\) (Sigma), which just means "sum of". Here are the three you need to know:

1. The sum of the first \(n\) integers (\(1 + 2 + 3 + \dots + n\)):
\(\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)\)

2. The sum of the first \(n\) squares (\(1^2 + 2^2 + 3^2 + \dots + n^2\)):
\(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\)

3. The sum of the first \(n\) cubes (\(1^3 + 2^3 + 3^3 + \dots + n^3\)):
\(\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2\)

Did you know? The sum of the cubes is actually just the square of the sum of the integers!
\(\sum r^3 = (\sum r)^2\). Pretty neat, right?

Top Tip: In the exam, the formulae for \(\sum r^2\) and \(\sum r^3\) are usually given in the formula booklet, but you must know how to use them algebraically to solve problems!

Key Takeaway: These formulae allow you to calculate huge sums instantly just by knowing the value of \(n\) (the number of terms).

3. The Method of Differences

This is a clever trick used to sum series that don't fit the standard formulae. The goal is to write each term in the series as a difference between two other things. When you add them all up, almost everything in the middle cancels out!

Analogy: Imagine a row of "telescoping" cups. When you push them together, they all collapse into each other, leaving only the very top and the very bottom. This is why this is sometimes called a Telescoping Series.

How to do it (Step-by-Step):

1. Express the general term \(u_r\) in the form \(f(r) - f(r+1)\).
2. Write out the first few terms and the last few terms of the sum.
3. Watch the "middle" terms cancel each other out!
4. Simplify the remaining terms (usually the first one or two and the last one or two).

Common Mistake: Don't rush the cancellation! Carefully write out \(r=1, r=2, r=n-1, r=n\) to see exactly which parts disappear.

4. Summation Using Partial Fractions

Wait, didn't we do Partial Fractions in standard Pure? Yes! And they are back. We use them here as a "gateway" to the Method of Differences.

If you see a series with a fraction like \(\sum \frac{1}{r(r+1)}\), you can't sum it as it is. But if you split it into:
\(\frac{1}{r} - \frac{1}{r+1}\)
...you can now use the Method of Differences to cancel everything out!

Quick Review Box:
- Split the fraction using A and B.
- Use the "Method of Differences" steps above.
- Enjoy the satisfying feeling of math terms vanishing!

5. Maclaurin Series: Approximating the "Hard Stuff"

Calculators are amazing, but how do they actually know the value of \(\sin(0.5)\) or \(e^{2.1}\)? They don't have a giant list of every possible answer. Instead, they use Maclaurin Series.

A Maclaurin series turns a complicated function (like a trig or exponential function) into an infinite polynomial (an endless string of \(x, x^2, x^3\dots\)).

The General Formula:

\(f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots + \frac{f^{(r)}(0)}{r!}x^r + \dots\)

Don't worry if this looks scary! It’s just a recipe:
1. Differentiate your function several times (\(f', f'', f'''\dots\)).
2. Plug \(x=0\) into the function and all its derivatives.
3. Put those numbers into the formula above.

Key Term: The General Term
The term \(\frac{f^{(r)}(0)}{r!}x^r\) is the general term. It tells you the pattern for any power of \(x\).

6. Standard Maclaurin Series to Memorise

The syllabus requires you to recognise and use these specific series. While they are in the formula book, being familiar with them will save you huge amounts of time.

1. Exponential: \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\) (Valid for all \(x\))
2. Sine: \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\) (Only odd powers!)
3. Cosine: \(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\) (Only even powers!)
4. Natural Log: \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots\) (Valid for \(-1 < x \le 1\))
5. Binomial: \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots\) (Valid for \(|x| < 1\))

Memory Aid:
- SIN is Strange (Odd powers).
- COS is Complimentary/Even (Even powers).
- Note that \(\ln(1+x)\) does NOT have factorials on the bottom, whereas the others do!

7. Validity and Convergence

Not every series works for every value of \(x\).
For example, if you try to use the \(\ln(1+x)\) series with \(x=10\), the numbers will just keep getting bigger and bigger—it won't give you a sensible answer. This is because the series only converges for a restricted set of values.

Key Takeaway: Always check the "validity" range. For \(\ln(1+x)\) and \((1+x)^n\), the "wiggle room" for \(x\) is quite small (usually between -1 and 1).

Summary Checklist

Before moving on to the next chapter, make sure you can:
- [ ] Use \(\sum r, \sum r^2, \sum r^3\) in algebraic proofs.
- [ ] Spot when to use the Method of Differences.
- [ ] Split a fraction and sum the series.
- [ ] Derive a Maclaurin series by differentiating.
- [ ] Use standard series to find approximate values (e.g., let \(x=0.1\) to find \(\sqrt{1.1}\)).