### Theory:

In this section, we will learn about geometric series.

Geometric series:
A geometric series is a series with all of its terms in a G.P.
Let us now try to find the sum of the $$n$$ terms in a G.P.

Let the terms in G.P be $$a$$, $$ar$$, $$ar^2$$, $$ar^3$$,$$…$$,$$ar^{n-1}$$.

$$S_n$$ $$=$$ $$a$$ $$+$$ $$ar$$ $$+$$ $$ar^2$$ $$+$$ $$ar^3$$ $$+...+$$ $$ar^{n - 1}$$ $$\longrightarrow (1)$$

On multiplying by $$r$$ on both the sides, we get:

$$rS_n$$ $$=$$ $$ar$$ $$+$$ $$ar^2$$ $$+$$ $$ar^3$$ $$+$$ $$ar^4$$ $$+...+$$ $$ar^{n}$$ $$\longrightarrow (2)$$

On subtracting $$(2)$$ from $$(1)$$, we get:

$$rS_n$$ $$-$$ $$S_n$$ $$=$$ $$ar^{n}$$ $$-$$ $$a$$

$$S_n$$$$(r - 1)$$ $$=$$ $$a$$$$(r^n -1)$$

$$S_n$$ $$=$$ $$\frac{a(r^n -1)}{r - 1}$$

Sum of $$n$$ terms in a series when $$r$$ $$=$$ $$1$$:

$$S_n$$ $$=$$ $$a$$ $$+$$ $$ar$$ $$+$$ $$ar^2$$ $$+$$ $$ar^3$$ $$+...+$$ $$ar^{n - 1}$$

$$S_n$$ $$=$$ $$a$$ $$+$$ $$a(1)$$ $$+$$ $$a(1)^2$$ $$+$$ $$a(1)^3$$ $$+...+$$ $$a(1)^{n - 1}$$

$$S_n$$ $$=$$ $$a$$ $$+$$ $$a$$ $$+$$ $$a$$ $$+$$ $$a$$ $$+...+$$ $$a$$

$$S_n$$ $$=$$ $$na$$

Sum of infinite terms in a series:

$$\text{Sum of infinite terms in a series}$$ $$=$$ $$a$$ $$+$$ $$ar$$ $$+$$ $$ar^2$$ $$+$$ $$ar^3$$ $$+...$$

$$\text{Sum of infinite terms in a series}$$ $$=$$ $$\frac{a}{1 - r}$$, $$-1 < r < 1$$