In this section, we explore the concept of Geometric Progressions (GPs), particularly focusing on their convergence and the sum to infinity. These concepts are fundamental in higher-level mathematics and are essential for understanding series and sequences.

Convergence in Geometric Progressions

• Definition and Criteria: A Geometric Progression (GP) is said to converge if the absolute value of its common ratio, $|r|$, is less than 1. This criterion ensures that each successive term in the GP decreases in magnitude, leading to a finite limit.
• Mathematical Insight: The convergence of a GP

• can be understood by examining the behavior of its terms. As the number of terms increases, if the absolute value of the common ratio is less than 1, the terms get progressively smaller and approach zero.
• Illustrative Example: Consider a GP with a common ratio of 0.5. As we progress through the terms, each term is half the previous term, leading to a diminishing sequence that converges.

Sum to Infinity in Convergent Geometric Progressions

• Fundamental Formula: The sum to infinity of a convergent GP is given by the formula $S_{\infty} = \frac{a}{1 - r}$, where 'a' is the first term and 'r' is the common ratio.
• Derivation and Understanding: This formula is derived under the premise that as the number of terms in the GP approaches infinity, the nth term becomes negligible. The sum of an infinite number of terms in a converging GP can be finite, and this formula encapsulates this concept.
• Practical Applications: This concept finds applications in various fields such as economics, physics, and engineering, where it is used to model and analyze scenarios involving infinite processes or series.

Examples

Example 1:

Calculate the sum to infinity of the GP: $2, 1, 0.5, \ldots$

Solution:

1. First term $(a)$ = 2, Common ratio $(r)$ = $\frac{1}{2}$ (second term divided by the first term).

2. The GP converges since $r = 0.5$ is less than 1.

3. Use the Sum to Infinity Formula: $S_{\infty} = \frac{a}{1 - r}.$

4. Calculate: $S_{\infty} = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4$.

Therefore, the sum to infinity of the GP $(2, 1, 0.5, \ldots)$ is 4.

Example 2:

Assess whether the GP $5, -2.5, 1.25, \ldots$ converges and find its sum to infinity if it does.

Solution:

1. First term $(a)$ = 5, Common ratio $(r)$ = $\frac{-2.5}{5} = -0.5$.

2. The GP converges since $|r| = |-0.5| = 0.5$ is less than 1.

3. Use the Sum to Infinity Formula: $S_{\infty} = \frac{a}{1 - r}$.

4. Calculate: $S_{\infty} = \frac{5}{1 - (-0.5)} = \frac{5}{1.5} = \frac{10}{3}$.

Therefore, the sum to infinity of the GP $(5, -2.5, 1.25, \ldots)$ is approximately 3.33.

Example 3:

Determine the convergence and sum to infinity of the GP $10, 20, 40, \ldots$

Solution:

1. First term $(a)$ = 10, Common ratio $(r)$ = $\frac{20}{10} = 2$.

2. The GP does not converge since the absolute value of the common ratio $|r| = |2| = 2$, is greater than 1.

3. A GP only has a finite sum to infinity if |r| < 1. As |r| > 1 in this case, the terms of the GP increase indefinitely.

Therefore, the GP $(10, 20, 40, \ldots)$ does not converge and thus does not have a finite sum to infinity.