91Ó°ÊÓ

Understanding a geometric series is crucial to solving the problem. A geometric series is a series where each term is a constant multiple (common ratio, r) of the previous term. For instance, in the series

• 1, r, r^2, r^3, ...

Each term is obtained by multiplying the previous term by r. In our exercise, the terms get more complex, becoming:

• 1, 2r, 3r^2, 4r^3, ...

Here, each term also has an increasing coefficient in front. The important property of geometric series is that it converges (i.e., adds up to a finite number) when the common ratio r is between 0 and 1 (0 < r < 1).

To better understand how the series converges, we need to consider its partial sum. A partial sum involves adding up the first n terms of the series. If we denote the partial sum by S_n, we can write it as:
S_n = 1 + 2r + 3r^2 + ... + nr^{n-1}.
Analyzing partial sums helps understand the behavior of the series as n increases. Comparing S_n with rS_n (which is the same sum but each term is multiplied by r) simplifies the calculation greatly.

A series converges if the sum of its infinite terms approaches a finite value. For the given series, we need to show that it converges and find its sum. To do this, we look at the series' behavior through its partial sums S_n. Using the difference between S_n and rS_n, we can derive an expression for S_n:
S_n(1-r) = 1 + r + r^2 + ... + r^{n-1} - nr^n.
The terms 1 + r + ... + r^{n-1} form a geometric series that sums to (1 - r^n) / (1 - r). Hence,
S_n(1-r) = (1 - r^n)/(1-r) - nr^n.
Isolating S_n gives us the formula to determine the sum. As n approaches infinity, the term r^n becomes negligible because 0 < r < 1. What's left is:
S = 1 / (1 - r)^2, showing that the series converges to this value.

In genetics, certain models use series to predict behaviors like the average lifespan of a harmful gene. Our series 1 + 2r + 3r^2 + ... describes such a genetic scenario over generations. Each term accounts for extended life by multiplying with r from previous life spans (which get smaller). This genetic model provides insight into how traits diminish or persist over time. By demonstrating the convergence of our series to 1/(1-r)^2, we show that these genetic traits eventually stabilize, giving scientists predictability in genetic variations involving harmful genes.