Question

Determine the radius of convergence of the given power series. $$ \sum_{n=0}^{\infty} 2^{n} x^{n} $$

Short answer

Answer: The radius of convergence for the given power series is 1/2.

Step-by-step solution

Write down the general term

The general term in the power series is: $$ a_n = 2^n x^n $$

Apply the Ratio Test

The Ratio Test states that if \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\), the series converges. Compute the ratio of consecutive terms: $$ \frac{a_{n+1}}{a_n} = \frac{2^{n+1} x^{n+1}}{2^n x^n} $$

Simplify the ratio

After simplifying the ratio, we have: $$ \frac{a_{n+1}}{a_n} = 2\cdot x $$

Apply the Ratio Test condition

As per the Ratio Test, we need to find the values of x for which: $$ \lim_{n\to\infty} \left| 2\cdot x \right| < 1 $$

Find the interval of x-values for convergence

To find the interval of x-values for which the series converges, we proceed as follows: $$ |2\cdot x| < 1 \implies -1 < 2\cdot x < 1 \implies -\frac{1}{2} < x < \frac{1}{2} $$

Calculate the radius of convergence

The radius of convergence is the size of the interval for which the series converges, divided by 2. In this case, the radius of convergence (R) is: $$ R = \frac{\frac{1}{2} - (-\frac{1}{2})}{2} = \frac{1}{2} $$ Thus, the radius of convergence for the given power series is \(\frac{1}{2}\).

Key concepts

Power Series

A power series is a fascinating mathematical concept that involves an infinite sum of terms, each of which is in the form of powers of a variable (often denoted as \( x \)). This is a bit like a polynomial, but it goes on infinitely. A power series generally looks like this:

• \( \sum_{n=0}^{\infty} a_n x^n \)
• Where \( a_n \) is the coefficient of the term, and \( x^n \) is the variable raised to the nth power.

In our example, the power series is \( \sum_{n=0}^{\infty} 2^{n} x^{n} \). Here, each term consists of \( 2^n \) as the coefficient, multiplied by \( x^n \), the variable raised to the nth power. Understanding power series is essential because they are used in many fields such as calculus, complex analysis, and even physics. They help approximate functions and solve differential equations. When dealing with them, it's crucial to determine where the series converges, which is exactly where the radius of convergence comes into play.

Ratio Test

The ratio test is a handy tool for determining whether a power series converges. It involves calculating the limit of the absolute ratio of successive terms in the series. Here’s what the process involves:

• First, write down the general term of the series, \( a_n \).
• Next, compute \( \frac{a_{n+1}}{a_n} \), the ratio of the \((n+1)\)th term to the nth term.
• Take the limit as \( n \) approaches infinity of the absolute value of this ratio.

If the limit is less than 1, the series will converge, meaning it reaches a finite sum. For the series \( \sum_{n=0}^{\infty} 2^{n} x^{n} \), we computed:

• \( \frac{a_{n+1}}{a_n} = 2 \cdot x \)

Given this expression, applying the ratio test means determining \( |2 \cdot x| < 1 \). This inequality helps us find the values of \( x \) for which the series converges, a critical step in evaluating convergence.

Convergence Interval

The convergence interval of a power series is the set of values for which the series converges. Deriving this interval is key to understanding the behavior of the power series. To find it, you follow these steps:

• Start with the condition from the ratio test: \( |2 \cdot x| < 1 \).
• Solving the inequality \( |2 \cdot x| < 1 \), gives the interval \(-\frac{1}{2} < x < \frac{1}{2} \).

This result shows that only the values of \( x \) between \(-\frac{1}{2}\) and \(\frac{1}{2}\) (not including \(-\frac{1}{2}\) and \(\frac{1}{2}\) themselves) will make the series converge. The convergence interval helps define the behavior and scope of the series, crucial for applications where precise function approximation is needed. Calculating this interval is vital for knowing where the power series "works" to approximate solutions effectively.