Question

Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty}(1.075)^{n} $$

Short answer

The given series \( \sum_{n=0}^{\infty}(1.075)^{n} \) diverges.

Step-by-step solution

- Identify the Common Ratio

The first step is to identify the common ratio of the given geometric series. The common ratio here is 1.075.

- Determine the Convergence or Divergence

Next, determine if the series converges or diverges. This is done by checking if the absolute value of the common ratio is less than 1 or not. For the series to converge, the absolute value of common ratio should be less than 1. Here, the absolute value of the common ratio, 1.075, is greater than 1.

- Conclude the Result

Based on the analysis in Step 2, since the absolute value of common ratio (1.075) is greater than 1, we can conclude that the given series \( \sum_{n=0}^{\infty}(1.075)^{n} \) diverges.

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series 1, 2, 4, 8, ..., each term is twice the previous term, which means the common ratio is 2.

Geometric series come up frequently in various branches of mathematics and have important applications in physics, engineering, biology, economics, and finance. Understanding whether a geometric series converges or diverges is crucial because it tells us whether the sum of all terms in the series approaches a finite value (converges) or grows without bound (diverges).

To test for convergence, we rely on the common ratio, and a simple criterion: if the absolute value of the common ratio is less than 1, then the geometric series converges; otherwise, it diverges. This is because when the ratio is less than one in absolute terms, successive terms in the series become increasingly smaller, drawing the sum closer to a finite limit.

Common Ratio

The common ratio is the factor by which consecutive terms of a geometric series are multiplied to obtain the next term. This number can be a fraction, an integer, or an irrational number, and it determines the behavior of the entire series. For example, in our given series, \( 1, 1.075, 1.1556, ... \), each term is 1.075 times the term before it, establishing 1.075 as the common ratio.

In order to determine if a series converges, one should first check whether the common ratio's absolute value is less than, equal to, or greater than one. If the common ratio is exactly 1, the series is constant and does not approach a limit; if it's greater than 1, the terms keep increasing to infinity, meaning the series diverges; if it's less than 1, each term gets smaller, implying that the series may converge to a limit.

Absolute Value

The absolute value of a number is a measure of its magnitude regardless of its sign. It's symbolized by two vertical bars on either side of the number, for instance, \( | -3 | = 3 \) and \( | 3 | = 3 \). When evaluating the convergence of a geometric series, the absolute value is of primary importance because it allows us to disregard the direction of the number (whether it's positive or negative) and focus solely on its size relative to one.

The concept of absolute value is used to enforce the convergence criteria of a geometric series. It doesn't matter if the common ratio is positive or negative; what is significant is its size when compared to the number one. For convergence, the absolute value of the common ratio must be strictly less than one; otherwise, the terms won't decrease in magnitude enough to approach a limit.