Q.Find the sum of the following series up to $n$ terms: $5+55+555+5555+\ldots$ OR If the A.M. and G.M. of two positive numbers $a$ and $b$ are $10$ and $8$ respectively, find the numbers.

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?

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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.

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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.

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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) …