Q.Find the sum of the series $.4 + .44 + .444 + \cdots$ to $n$ terms. OR Show that the ratio of the sum of first $n$ terms of a G.P to the sum of the terms from $(n+1)^{th}$ to $(2n)^{th}$ terms is $\dfrac{1}{r^n}$.

What Does "Sum of a Series" Even Mean?

Imagine you're standing at point 0 and you take a step of 1 metre forward. Then a step of half a metre. Then a quarter metre. Then an eighth. And you keep going, each step half the size of the previous one.

After 1 step: you're at 1 metre.

After 2 steps: at 1.5 metres.

After 3 steps: at 1.75 metres.

After 4 steps: at 1.875 metres.

After 5 steps: at 1.9375 metres.

You notice something: you're getting closer and closer to 2 metres, but you never quite reach it. If you could take infinitely many steps, would you ever get to exactly 2 metres? This is the heart of what a series is — adding up infinitely many numbers and asking: does this sum settle down to a finite value?

The Precise Definition

Let’s formalise this. Suppose we have an infinite sequence of numbers:

We want to make sense of the infinite sum:

We can't just "add infinitely many things" directly — that's not a finite operation. So mathematicians do something clever: they look at partial sums.

Define:

$S_n$ is called the $n$th partial sum — it's the sum of the first $n$ terms.

Now, the series is said to converge (or have a sum) if the sequence of partial sums $S_1, S_2, S_3, \dots$ approaches some finite number $S$ as $n$ gets larger and larger. In that case, we write:

If the partial sums don't settle down to a finite number — they either grow without bound or oscillate forever — the series diverges and has no finite sum.

The Walking Example, Formalised

Our sequence of steps: $1, \frac12, \frac14, \frac18, \dots$

The partial sums:

You can prove (and we will later) that:

As $n \to \infty$, $\frac{1}{2^{n-1}} \to 0$, so $S_n \to 2$. Therefore:

The infinite sum equals exactly 2 — even though you never "reach" it after any finite number of steps, the limit of the process is 2.

Two Classic Examples to Build Intuition

1. The Harmonic Series (Diverges)

This one is tricky. The terms get smaller and smaller, but the partial sums grow without bound — just very slowly. $S_{100} \approx 5.18$, $S_{1000} \approx 7.48$, $S_{1000000} \approx 14.39$. It never stops growing. This series diverges.

2. A Geometric Series (Converges) …