Question

Is \(\sum_{i=0}^{\infty}(5 x-20)^{k}\) a power series? If so, find the center \(a\) of the power series and state a formula for the coefficients \(c_{k}\) of the power series.

Short answer

Question: Determine if the given expression is a power series, and if so, find its center and provide a formula for the coefficients. The given expression is ∑(5x-20)^k. Answer: The given expression is a power series with a center at a = -20 and coefficients of c_k = 1 for all values of k.

Step-by-step solution

Determine if the expression is a power series

A power series is an infinite series of the form \(\sum_{k=0}^{\infty}c_k(x-a)^k\). Comparing this definition to the given expression, \(\sum_{i=0}^{\infty}(5x-20)^{k}\), we can see that it indeed has the form of a power series.

Identify the center of the power series

We can determine the center of the power series by comparing the given expression with the standard form of a power series, \(\sum_{k=0}^{\infty}c_k(x-a)^k\). In our case, the given expression is \((5x-20)^k\), and the corresponding general term in the standard power series is \((x-a)^k\). Equating the two expressions, we have: \((5x-20)^k = (x-a)^k\) Since the exponents are equal, we can simplify the equation by comparing the bases: \(5x-20 = x-a\) Now, we solve for \(a\): \(a = 5x-x-20\) \(a= 4x-20\) Since the center \(a\) should be a constant, we solve for \(x\) when \(a = 4x - 20\): \(4x - 20 = a\) Adding 20 to both sides: \(4x = a+20\) Dividing by 4: \(x = \frac{a + 20}{4}\) Now, to find the value of \(a\) that makes the right side a constant, we set the numerator equal to zero: \(a + 20 = 0\) Solving for \(a\): \(a = -20\) So, the center of the power series is \(a = -20\).

Identify a formula for the coefficients \(c_k\)

Now, we know that \((5x-20)^k =(x-a)^k = (x - (-20))^k = (x + 20)^k\). Thus, we can see that the coefficients \(c_k\) in our given expression are all equal to 1, as there is no additional multiplier in the expression with the power of \(k\). Therefore, the formula for the coefficients is simply: \(c_k = 1\) for all values of \(k\). #Conclusion# In conclusion, the given expression \(\sum_{i=0}^{\infty}(5x-20)^{k}\) is a power series with a center at \(a = -20\) and coefficients of \(c_k = 1\) for all values of \(k\).

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides a framework for modeling systems in which there is change and a way to deduce the predictions of such models. One fundamental aspect of calculus is the concept of a limit, which leads to the definitions of derivatives and integrals. These concepts are used to analyze the behavior of functions with respect to their variables.

In the context of the power series problem, calculus helps us understand how a function can be expressed as an infinite sum of its derivatives at a certain point, which is the essence of a Taylor series. In the given exercise, identifying the power series and determining the center and the coefficients within it are part of the broader scope of calculus, where we effectively dissect the function into a form that can be more easily analyzed and manipulated.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It is written in the form \(\sum_{n=a}^{\infty} u_n\), where \(u_n\) is the nth term of the sequence. The series can be of various types depending on the nature of the terms, such as arithmetic series, geometric series, and power series.

The given exercise deals with a power series, that is a special kind of infinite series where each term is a product of a constant coefficient \(c_k\) and a power of \(x - a\). Understanding infinite series is crucial for many areas of mathematics and physics, as they can represent complex functions and phenomena through a more simplified infinite sum of predictable and understandable terms.

Series Convergence

Series convergence is about determining whether an infinite series approaches a certain value, known as its sum, as the number of terms increases. For a series to be convergent, the sequence of partial sums \(\sum_{n=a}^{N} u_n\), for \(N\) approaching infinity, should approach a finite limit. If a series does not converge to a limit, it’s said to diverge.

Convergence tests, such as the Ratio Test, Root Test, and Integral Test, are used to determine whether a series converges or diverges. When dealing with power series, the radius of convergence is an essential concept. It defines the interval around the center \(a\) within which the series converges. Locating the center of the power series, as done in the exercise, is the first step towards understanding the series' behavior around that point and whether it converges.