Question

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{2 k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Short answer

Question: Determine the values of the function \(f(x) = \sum_{k=0}^{\infty} x^{2 k}\) at the points 0, 0.2, 0.5, 1, and 1.5. Also, find the domain of this function. Answer: The values of the function are as follows: - \(f(0) = 1\) - \(f(0.2) \approx 1.0417\) - \(f(0.5) \approx 1.3333\) - \(f(1)\) is undefined - \(f(1.5) = -0.8\) The domain of \(f(x)\) is \((-1, 1)\).

Step-by-step solution

Finding the general expression of the geometric series

The given geometric series: \(f(x) = \sum_{k=0}^{\infty} x^{2 k}\) Let's find the expression for this sum. For a geometric series to converge, the absolute value of its common ratio must be less than 1 (|r|<1). In this case, the common ratio is \(x^2\). Thus, for convergence: \(|x^2| < 1\) Now, the sum of an infinite geometric series is given by the formula: \(\frac{a}{1-r}\) Here, \(a\) is the first term and \(r\) is the common ratio. For our series, \(a=1\) (the first term, when k=0) and \(r=x^2\). So, the expression for \(f(x)\) is: \(f(x) = \frac{1}{1-x^2}\) Now we can use this expression to find the values of \(f(x)\) at the given points.

Evaluating \(f(x)\) at given points

Using the expression we found in Step 1, evaluate \(f(x)\) at each given point: 1. \(f(0) = \frac{1}{1-0^2} = \frac{1}{1} = 1\) 2. \(f(0.2) = \frac{1}{1-(0.2)^2} = \frac{1}{1-0.04} = \frac{1}{0.96} \approx 1.0417\) 3. \(f(0.5) = \frac{1}{1-(0.5)^2} = \frac{1}{1-0.25} = \frac{1}{0.75} \approx 1.3333\) 4. \(f(1) = \frac{1}{1-(1)^2} = \frac{1}{1-1}\), which is undefined since division by zero is not allowed. 5. \(f(1.5) = \frac{1}{1-(1.5)^2} = \frac{1}{1-2.25} = \frac{1}{-1.25} = -0.8\) Results: - \(f(0) = 1\) - \(f(0.2) \approx 1.0417\) - \(f(0.5) \approx 1.3333\) - \(f(1)\) is undefined - \(f(1.5) = -0.8\)

Finding the domain of \(f(x)\)

The domain of \(f(x)\) is the set of all \(x\) values for which the geometric series converges. As mentioned earlier, the series converges when: \(|x^2| < 1\) We can solve this inequality: \(-1 < x^2 < 1\) Taking the square root of each term (and noting that the inequality signs stay the same since all terms are positive): \(-\sqrt{1} < x < \sqrt{1}\) Thus, the domain of \(f(x)\) is \(-1 < x < 1\) or in interval notation: Domain of \(f(x) = (-1, 1)\)

Key concepts

Convergence of Geometric Series

Understanding the convergence of a geometric series is crucial in mathematics, particularly in calculus and series analysis. 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. The series converges when the absolute value of this common ratio is less than one, e.g., |r| < 1. This condition ensures that as we proceed to infinity, the terms of the series get smaller and approach zero.

For instance, consider the series f(x) = \( \frac{1}{1 - r} \), with the common ratio being x^2. The series converges only if |x^2| < 1, meaning that x must be between -1 and 1, exclusive. An intuitive way of visualizing this constraint is considering that for the sum to stabilize to a finite value, the addition of infinite ever-smaller terms must not reach an unbounded total. When x falls outside this interval, the terms do not diminish adequately, leading the series to diverge or, in some cases such as x=1, it isn't defined at all.

Domain of a Function

In many mathematical problems, identifying the domain, which consists of all the input values (usually x values) for which the function is defined, is crucial for understanding the behavior of the function. The domain is influenced by factors such as division by zero and taking square roots of negative numbers, which are not allowed in real number calculations.

For the function f(x) = \( \frac{1}{1 - x^2} \), the domain excludes values which would make the denominator zero, i.e., x = \pm 1. As a result, this constraint narrows down the domain of f(x) to values strictly between -1 and 1, written as (-1, 1). Understanding the domain is not just a technicality but a crucial aspect that dictates the applicability of the function and, consequently, the ability to process and interpret the function's output correctly.

Sum of an Infinite Geometric Series

The sum of an infinite geometric series is a fascinating concept illustrating both the beauty and utility of mathematics. When a geometric series has a convergence criterion satisfied, which means its common ratio |r| < 1, it can be summed to a finite value using the formula S = \( \frac{a}{1 - r} \), where S is the sum, a is the first term, and r is the common ratio.

In the series f(x)=\sum_{k=0}^{\infty} x^{2k}, the sum can be found (for values of x within the domain) by plugging into this formula, yielding f(x) = \( \frac{1}{1 - x^2} \), as the first term a is 1 and the common ratio r is x^2. This results in a simple yet powerful expression that not only defines the series in a concise way but also gives us a precise value for the sum when x is within the convergence interval.

Calculus

Calculus, a branch of mathematics, offers tools to analyze changes and motion, often dealing with concepts of limits and infinite processes. The fundamental theorems of calculus bridge the concepts of derivatives (which represent rates of change) and integrals (which represent accumulation of values).

When it comes to series and particularly geometric series, concepts from calculus are applied to determine convergence and evaluate sums. Calculus provides a formal framework to work with infinite processes, like infinite series, by allowing us to grasp how adding infinitely many terms can lead to a finite result. This counterintuitive outcome is crucial for solving advanced problems in physics, engineering, and economics, where such summations frequently appear. In context, the function f(x) developed through calculus by summing a geometric series can be used to model growth processes, investment returns, or waveforms in physics, illustrating the practical and far-reaching applications of calculus.