A geometric progression is a progression where each term is obtained by multiplying a constant number, called the common ratio, by the preceding term.
Some examples of Geometric Progression:
As you can see that in geometric progressions, the terms and the common ratio can indeed be any number, including complex, irrational, and real numbers.
Let us write the progression in its algebraic form, where \(a_n\) denotes the nth term, \(r\) is the common ratio, and \(S_n\) is the sum of the first n terms of a geometric series.
\[a_1,a_1r,a_1r^2,a_1r^3,\cdots\]An nth term in a geometric progression is written as
Thus, what we mean by the sum of the first n terms of a geometric progression is
Multiplying both sides by r gives us,
Putting equation (2) in (3) we get,
\[rS_n=S_n-a_1+a_1r^n \implies (r-1)S_n=a_1(r^n-1)\]Gives us our final equation
\[S_n=a_1\frac{(r^n-1)}{(r-1)}\]This derivation is with the use of summation. Those not familiar with the concept may skip.
The sum of the first n terms of a geometric progression is
\[S_n=\sum_{i=1}^{n}a_i=\sum_{i=1}^{n}a_1r^{i-1}\]We know, using the binomial expansion
\[(a^n-b^n)=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+ab^{n-2}+b^{n-1})\]Putting a=1 and b=r,
Using equation (5) in (4) , we can easily write,
\[S_n=a_1\frac{(1-r^n)}{(1-r)}\]Individuals who are not acquainted with the concept of infinity are advised to become familiar with this concept before further engagement.
An infinite geometric series is characterized by its continuous nature, wherein each term is derived by multiplying the preceding term by a constant known as the common ratio.
A general infinite geometric series can be written as
\[a,ar,ar^2,ar^3,ar^4,\cdots\]And the sum of this infinite series can be written as,
\[S=a+ar+ar^2+ar^3+ar^4+ar^5+\cdots =\sum_{n=0}^{\infty}ar^n\] \[S=a+r(a+ar+ar^2+ar^3+\cdots)=a+rS\]In light of thisoutcome, we will analyze the behavior of this formula.
\(\text{ Case 1: }\) When \(a > 0\) and \(r > 0\) , the sum will consistently yield a positive value. However, the equation (6) tells us that if \(r > 1\), then \(S < 0\). This presents a contradiction (because we can easily see in the series that all its terms are positive, and adding an infinite number of positive number always reults in being a positive number. Though saying this contradicts a very famous infinite series that we are going to talk about in the Interesting Infinite Series ). Therefore, we establish the constraint \(r < 1\) in order for the formula to yield a meaningful and accurate outcome.
\(\text{ Case 2: }\) When \(a>0\) and \(r<0\) , this should be dealt by further diving into two cases(as you will see later why we did that?). To deal with the following cases, we break the series into two parts as follows: From equation (5) and (6) we can see if
\[S=a+ar+ar^2+ar^3+\cdots\] \[\implies S=(a+ar^2+ar^4+\cdots)+(ar+ar^3+ar^5+\cdots)\]Let’s \(S_1\) and \(S_2\) be \((a+ar^2+ar^4+\cdots)\) and \((ar+ar^3+ar^5+\cdots)\) respectively.
Thus S becomes \(S=S_1+S_2\), where
\[\label{eq:enumeration5} S_1=\frac{a}{1-r^2}\]and
\[\label{eq:enumeration6} S_2=\frac{ar}{1-r^2}\]\(\text{ Case 2A: }\) When \(-1<r<0\), the formula tells that \(S_1>0\), \(S_2<0\) and \(S>0\). which implies that \(|S_1|>|S_2|\).
(Notice that we have taken the absolute value, not the actual value of \(S_1\) and \(S_2\) to avoid further confusion.)
Comparing the absolute value of equation (5) and (6) , one can easily confirm that this happens when \(|r|<1\).
And also looking at the behavior of the series expansion of \(S_1\) and \(S_2\) we can confirm that in this case this formula holds as the formula says that \(|S_1|>|S_2|\) and comparing each term of the series expansion of \(S_1\) and \(S_2\) ,we get, since we know that when any number less than 1 and greater than -1 is raised to some positive power its absolute value always decreases. Thus, we can say by comparing each term that \(|S_1|>|S_2|\).
\(\text{ Case 2B: }\) When \(r<-1\) , the formula tells that \(S_1<0\) , \(S_2>0\) and \(S>0\) . which implies that \(|S_1|<|S_2|\).
But when we look at the series expansion of \(S_1\) . As all the terms are raised to an even power, thus \(S_1>0\) . A contradiction.
We will see the convergence and divergence proof in the section on A General Talk on Sequences and Series
A pertinent question may have arisen in the reader’s mind. Initially, when we began deriving this equation, we stated, "Look, here is an equation; let us find the general formula for its sum." However, upon reading up the section on sequences and series the reader will get to know that it is essential to first ascertain whether the series in question is convergent or divergent.
It’s a common misconception that mathematics is just a collection of rigid formulas. So when we discover a situation where a formula fails, it’s tempting to call it an 'exception'—a frustrating rule to be memorized, much like we might encounter in an empirical science.
But the beauty of mathematics is that there are no true exceptions.
When a mathematical formula breaks down, it isn’t an arbitrary quirk. Instead, it’s a logical signpost that we’ve reached the boundary of a concept. Exploring why a formula fails in certain cases doesn't lead to a list of exceptions; it leads to a deeper understanding, revealing the fundamental reasoning and elegant patterns that define the mathematical world.
The formula exhibits limitations at particular points along the number line due to the convergence characteristics of the series. A comprehensive proof will be provided in the section titled "A General Talk on Sequences and Series"
Individuals who are not acquainted with the concept of limits may opt to disregard this additional proof.
Consider a geometric series with first term \(a\) and common ratio \(r\) . The sum of the first \(n\) terms is given by:
\[S_n = a + ar + ar^2 + \cdots + ar^{n-1}.\]This is a finite geometric series, and it has the closed-form expression:
\[S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for } r \neq 1.\]Now, consider the infinite series:
\[S = \lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{a(1 - r^n)}{1 - r}.\]We analyze the behavior of the term \(r^n\) as \(n \to \infty\):
\(\text{ Case 1: }\) If \(|r| < 1\) , then
\[\lim_{n \to \infty} r^n = 0\]So the sum becomes:
\[S = \frac{a(1 - 0)}{1 - r} = \frac{a}{1 - r}, \quad \text{(Converges )}.\]\(\text{ Case 2: }\) If \(|r| > 1\) , then
\[\lim_{n \to \infty} r^n = \infty\]and so \(S_n \to \infty\).
The series diverges.
\(\text{ Case 3: }\) If \(r = -1\) , then \(r^n\) alternates between \(1\) and \(-1\) , and does not approach a single value. Thus, \(S_n\) does not converge to a finite limit (oscillates).
\(\text{ Conclusion: }\)
\[\boxed{ \text{The infinite geometric series converges if and only if } |r| < 1.}\]