Problem 3 - Advanced Engineering Mathematics

Question

In Problems \(1-4\), find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=1}^{\infty} \frac{(-1)^{k}}{\underline{10^{k}}}(x-5)^{k} $$

Step-by-Step Solution

Answer

Radius: 10; Interval: \((-5, 15)\).

Step 1: Identify the General Form of the Power Series

The given power series is \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{10^{k}}(x-5)^{k}\). It matches the general form of a power series \(\sum_{k=0}^{\infty} a_k (x-c)^k\), where \(a_k = \frac{(-1)^{k}}{10^{k}}\) and \(c = 5\).

Step 2: Use the Ratio Test

To find the radius of convergence \(R\), use the Ratio Test: \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|.\) Substitute \(a_k = \frac{(-1)^{k}}{10^{k}}\) into the test.

Step 3: Calculate the Ratio

Substitute the terms into the formula: \(\frac{a_{k+1}}{a_k} = \frac{((-1)^{k+1}/10^{k+1})}{((-1)^{k}/10^{k})} = \frac{-1}{10}.\) Take the absolute value: \(\left|\frac{-1}{10}\right| = \frac{1}{10}.\)

Step 4: Solve for Radius of Convergence

The limit calculated from the Ratio Test is \(\frac{1}{10}\). Since the Ratio Test gives \(\left|\frac{a_{k+1}}{a_k}\right| < 1\), we find the radius of convergence: \(R = 10\).

Step 5: Determine Interval of Convergence

For \(||x-5|| < 10\), solve the inequality: \[-10 < x - 5 < 10\] Add 5 to all sides: \[-5 < x < 15\].

Step 6: Evaluate Endpoints

Test endpoints \(x = -5\) and \(x = 15\) in the series to check for convergence: - At \(x = -5\), the series becomes \(\sum_{k=1}^{\infty}\frac{(-1)^{k}((-10)^{k})}{10^{k}} = \sum_{k=1}^{\infty} (-1)^k\), which diverges.- At \(x = 15\), the series becomes \(\sum_{k=1}^{\infty}(-1)^{k}(1) = \sum_{k=1}^{\infty} (-1)^k\), which also diverges.Thus, the interval of convergence is \((-5, 15)\).

Key Concepts

radius of convergence

In the world of power series, the radius of convergence is a key concept. It tells us the distance from the center of the series, within which the series converges. For our given power series, the center is at 5 because of the \(x-5\) term.

Now, to find the radius of convergence, we often use the **Ratio Test** which involves looking at the sequence of coefficients. When we applied the Ratio Test to this particular series, we discovered that the limit of the absolute value of the ratio of successive terms was \(\frac{1}{10}\).

This result tells us that the series converges when the distance \(x-5\) is less than 10 units away from the center, giving us a radius of convergence \(R = 10\).

This doesn't mean the series converges uniformly everywhere within this radius, but rather it's a threshold up to which we can be sure that the series converges.

interval of convergence

Finding the interval of convergence involves determining where the series converges, not just the radius. Once we've established that the radius of convergence is 10, we must determine the interval on the number line where the series will indeed converge.

Because the radius is 10, the basic interval we start with is \(-10 < x - 5 < 10\). By solving this inequality, we discover \(-5 < x < 15\) as the potential interval.

But we must check the endpoints, \(x = -5\) and \(x = 15\), for convergence as sometimes the series could still be divergent at these points. In our given series, testing both endpoints revealed divergence at each. Therefore, they are not included in the interval of convergence.

Thus, the final interval of convergence for our series turned out to be \((-5, 15)\) without including the endpoints.

ratio test

The Ratio Test is a common technique used to determine the radius of convergence for a power series. It helps to analyze a series by checking the behavior of the sequence of its terms.

For our series, \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{10^{k}}(x-5)^{k}\), the Ratio Test involves calculating the ratio of consecutive terms. This process is summarized by finding \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\.

By substituting the terms \(a_k = \frac{(-1)^{k}}{10^{k}}\) and solving, we found that \left|\frac{-1}{10} \right| = \frac{1}{10}\. This gives us a clear indication of the series' convergence characteristics.

• If the result is less than 1, the series converges.
• If it is greater than 1, the series diverges.
• If it equals 1, the test is inconclusive.

In this case, with the result being less than 1, the series converges within a certain radius.

power series

A power series is a series of terms that involves powers of a variable, typically expressed in the form \sum_{k=0}^{\infty} a_k (x-c)^k\.

The given power series \sum_{k=1}^{\infty} \frac{(-1)^{k}}{10^{k}}(x-5)^{k}\ represents a power series centered at \(c = 5\) with coefficients \(a_k = \frac{(-1)^{k}}{10^{k}}\).

• The beauty of a power series lies in its ability to represent functions with an infinite sum of terms, each contributing to the behavior of the whole.
• It can resemble many functions such as polynomials, exponential functions, and trigonometric functions.
• Power series are extremely useful in mathematical analysis because they enable approximation of functions to an arbitrary degree of accuracy within their interval of convergence.

They serve as a cornerstone in calculus and analysis, providing deep insights into function behavior and enabling mathematicians to manage infinite processes.