Question

A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} \frac{(x+4)^{n}}{n !}$$

Short answer

Radius: \( \infty \); Interval: \((-\infty, \infty)\).

Step-by-step solution

Identify the Series

We are given a power series: \( \sum_{n=0}^{\infty} \frac{(x+4)^{n}}{n!} \). This resembles the standard form of a series \( \sum_{n=0}^{\infty} c_n (x-a)^n \) where \( c_n = \frac{1}{n!} \) and \( a = -4 \).

Apply the Ratio Test

To find the radius of convergence, apply the Ratio Test. Consider the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \) where \( a_n = \frac{(x+4)^n}{n!} \). This gives us: \[ \left| \frac{(x+4)^{n+1}}{(n+1)!} \cdot \frac{n!}{(x+4)^n} \right| = \left| \frac{x+4}{n+1} \right| \].

Determine the Limit

Evaluate the limit as \( n \to \infty \) of \( \left| \frac{x+4}{n+1} \right| \): \[ \lim_{n \to \infty} \left| \frac{x+4}{n+1} \right| = 0 \]. This limit is always 0, indicating that the series converges for all \( x \).

Conclude Radius of Convergence

Since the limit is 0 for any \( x \), the radius of convergence is \( R = \infty \). Thus, the series converges for all real numbers \( x \).

Interval of Convergence

Given that the radius of convergence is \( R = \infty \), the interval of convergence is \( (-\infty, \infty) \), meaning the series converges for every real number.

Key concepts

Interval of Convergence

The Interval of Convergence is essentially the set of all values that make a series converge. When studying power series, we often want to find out which input values, particularly the values of the variable \( x \), result in a convergent series. The interval can be finite or infinite.

• If the interval is finite, it means there are specific boundary values where the series might start diverging.
• An infinite interval of convergence, like \((-\infty, \infty)\), suggests the series converges for every real number.

In our exercise, we find the interval of convergence by first determining the radius of convergence using the ratio test. As it turned out, the series converges for all real numbers, giving us an interval of convergence of \((-\infty, \infty)\). This essentially means there are no restrictions on \( x \) for the series to remain convergent.

Ratio Test

The Ratio Test is a useful tool to determine the convergence or divergence of a series. It involves analyzing the behavior of the ratio of successive terms as the number of terms, \( n \), approaches infinity. For a given series \( \sum a_n \), we compute the absolute value of the ratio: \[\left| \frac{a_{n+1}}{a_n} \right|\]

• If the limit of this ratio is less than 1, the series converges.
• If the limit is greater than 1, or is infinite, the series diverges.
• If the limit equals 1, the test is inconclusive.

In the case of our power series, applying the Ratio Test involved calculating:\[\lim_{n \to \infty} \left| \frac{x+4}{n+1} \right|\]This limit equals 0 for any real \( x \), clearly indicating convergence. Therefore, according to the Ratio Test, our series converges for all \( x \), hinting at an infinite radius of convergence.

Power Series

A Power Series is a type of series where each term is a function of a variable raised to a power, typically expressed as:\[\sum_{n=0}^{\infty} c_n (x-a)^n\]

• Here, \( c_n \) is the coefficient of the series, \( x \) is the variable, and \( a \) is the center of the series.
• A power series can often converge for a range of \( x \) values around \( a \).

In our exercise, the given power series was:\[\sum_{n=0}^{\infty} \frac{(x+4)^n}{n!}\]Here, the series is centered at \( x = -4 \). Power series are quite versatile and can be used in various applications including solving differential equations and representing functions as infinite sums.

Convergence of Series

Convergence of Series refers to whether the sum of the terms in a series approaches a finite limit as the number of terms increases. This is different from a sequence, where the focus is on the terms themselves rather than their summation.

• A series converges if the partial sums of the series approach a specific number as we add more terms.
• If the partial sums do not approach a specific value, the series diverges.

In the context of power series, which are infinite series, determining convergence often depends on whether the series converges for particular values of \( x \). The Ratio Test and other techniques help determine this.
In our given power series:\[\sum_{n=0}^{\infty} \frac{(x+4)^n}{n!}\]We found the series converges for all real \( x \), as concluded through our findings regarding the interval of convergence and the ratio test. This confirms that the series' terms sum up to a finite number for any real value of \( x \).