Question

What is the formula for the sum of an infinite geometric series?

Answer 1

Hint: A sequence of non-zero numbers is called a geometric progression, if the ratio of a term and the term preceding it is always a constant quantity. That ratio is called the common ratio of the G.P. We will derive the formula for the sum of n terms of a geometric progression and then in that formula we will apply the limit of n tending to infinity.

Complete answer:
Let \[{S_n}\] denote the sum of n terms of a G.P with first term \[a\] and common ratio \[r\]. Then we can write as;
\[{S_n} = a + ar + a{r^2} + ...... + a{r^{n - 1}}.......(1)\]
Now multiplying both sides by \[r\], we get;
\[r{S_n} = ar + a{r^2} + ...... + a{r^{n - 1}} + a{r^n}......(2)\]
Subtracting \[(2)\] from \[(1)\] we get;
\[{S_n} - r{S_n} = a - a{r^n}\]
Because the rest all terms are common in both and will cancel each other.
Taking common we get;
\[ \Rightarrow {S_n}\left( {1 - r} \right) = a\left( {1 - {r^n}} \right)\]
On shifting we get;
\[ \Rightarrow {S_n} = \dfrac{{a\left( {1 - {r^n}} \right)}}{{\left( {1 - r} \right)}}\]
Here, \[r\] should not be equal to \[1\].
Now this is the sum of n terms of a GP.
\[ \Rightarrow {S_n} = \dfrac{a}{{1 - r}} - \dfrac{{a{r^n}}}{{1 - r}}\]
Now suppose \[|r| < 1\] i.e., \[ - 1 < r < 1\].
Now as \[ - 1 < r < 1\], so, if we increase the power of \[r\], then its value will decrease. When we put the power \[n \to \infty \], then \[{r^n} \to 0\].
Let the sum of infinite terms of this GP be \[S\]. So, we get;
\[ \Rightarrow S = \mathop {\lim }\limits_{n \to \infty } {S_n}\]
Putting the value, we get;
\[ \Rightarrow S = \mathop {\lim }\limits_{n \to \infty } \left( {\dfrac{a}{{1 - r}} - \dfrac{{a{r^n}}}{{1 - r}}} \right)\]
Now we know, \[n \to \infty \], then \[{r^n} \to 0\] because \[ - 1 < r < 1\].
So, we get;
\[ \Rightarrow S = \mathop {\lim }\limits_{n \to \infty } \left( {\dfrac{a}{{1 - r}} - \dfrac{{a \times 0}}{{1 - r}}} \right)\]
On simplification we get;
\[ \Rightarrow S = \dfrac{a}{{1 - r}}\], \[ - 1 < r < 1\].

Note:
A convergent sequence is the one which has a finite and unique value. The geometric progression will be convergent only when \[|r| < 1\]. This means we will get a finite and unique value for the infinite series only when \[|r| < 1\]. That is why we have applied for this case. If this is not the case then, the series will be divergent and we will not get a finite value.