By Aadi Mankodi, Mathematics BSc student,
‘God created the integers; all else is the work of man”
Leopold Kronecker
Hi – I’m Aadi, a second-year mathematics undergraduate at King’s College London. In this article I’d like to provide an insight into “Sequences & Series”, my favourite first-year maths module.
When I arrived at my first lecture, sitting in the Bush House auditorium, I expected it to be full of Greek letters and abstract concepts.
It wasn’t.
Instead, it began with something far simpler: definitions.
Starting From First Principles
We began with ideas that look almost too basic to matter:
• A set is a collection of numbers.
• The natural numbers, ℕ = {1,2,3,..}.
• A function f: X → Y assigns each element in X to exactly one element in Y.
• A sequence is a function Sn : ℕ → ℝ
Convergence: When Intuition Ceases to Suffice
From A-level, we’re allowed to use statements such as “as n → ∞ 1 → 0
n without proof. And intuitively, it feels rather obvious:
1, ½ 1/3, ¼ 1/5 ,…
The numbers get smaller.
And smaller.
And smaller.
But university mathematics asks the question: ”well, how can we formally prove it?”.
Here is the rigorous definition for what is means for a sequence to converge.
In words: if you choose how close you want your sequence to be to the limit (𝜖), then I ‘ll find a point (N) after which all terms stay that close.
Divergence
Not all sequences behave so nicely.
If a sequence does not converge, it is said to diverge.
Examples are:
• Sn= n
• Sn= {-l}n
• Sn= sin(n)
Divergence to infinity has a neat, precise definition:
In other words, ”you give me any positive number H; I ’11 find a point beyond which the sequence escapes it”. Makes sense, right?
Mathematics at King’s
Sequences & Series perfectly encapsulates what mathematics at King’s College London is all about. Concepts are carefully built from first principles. Every subtlety is addressed, not rushed over. And in doing so, you end up with a deeper understanding of the mathematics.
Theory can only take you so far. To practice, you are taught in tutorials: classes of around 15-20 students, where you work together to discuss the week’s problem sheet.
I’ve found office hours particularly valuable. Being able to sit down with a lecturer and ask questions makes a real difference.
Of course, as you progress in your degree, you can choose to specialize in areas that interest you. For me, that’s probability. Perhaps you’ll enjoy physics, finance, or number theory! Regardless, King’s has an enthusiastic community of lecturers and expert PhD students, who are often at the forefront of research in their own fields.
First-year mathematics isn’t about memorization. It’s about learning how to think.
In principle, anyone could master maths, provided they’re willing to slow down, be patient, and build understanding step by step.
And, of course, provided they can count.
Aadi
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