Problem 16
Question
We identify \(a=4 i\) and \(z=1 / 3 .\) since \(|z|=1 / 3<1\) the series converges to \\[\frac{4 i}{1-1 / 3}=6 i\\]
Step-by-Step Solution
Verified Answer
The series converges to \(6i\).
1Step 1: Determine the convergence condition
The series converges if the absolute value \(|z| < 1\). Given that \(|z|=1/3\), the condition is satisfied because \(1/3 < 1\). Therefore, the series converges.
2Step 2: Apply the Formula for the Sum of a Convergent Series
For a convergent geometric series with initial term \(a\) and common ratio \(z\), the sum is given by \(S = \frac{a}{1-z}\). Here, \(a = 4i\) and \(z = 1/3\).
3Step 3: Substitute Values into the Formula
Substitute \(a = 4i\) and \(z = 1/3\) into the formula: \(S = \frac{4i}{1 - 1/3}\).
4Step 4: Simplify the Denominator
Calculate the denominator \(1 - 1/3\): \[1 - 1/3 = 2/3.\]
5Step 5: Perform the Final Calculation
Replace the denominator in the formula and simplify the expression: \[S = \frac{4i}{2/3} = 4i imes \frac{3}{2} = 6i.\]
Key Concepts
Convergence of SeriesComplex NumbersSum of Series
Convergence of Series
In mathematics, especially in calculus and analysis, the concept of convergence is crucial when dealing with series. A series is simply an infinite sum of terms. For these series to converge, their terms must approach zero as you progress through them. When the terms are added infinitely, the total must approach a specific value, rather than growing indefinitely.
For geometric series, which include terms derived from multiplying a starting value by a constant ratio, we have a straightforward criterion for convergence. The series converges if the absolute value of the common ratio \(z\) is less than one, denoted as \(|z| < 1\). This ensures that as you add more terms, their impact on the sum becomes smaller and smaller, eventually stabilizing to a limit.
In the given exercise, the common ratio \(z = \frac{1}{3}\) meets this requirement, confirming that the series converges. Understanding convergence is key to practicing and applying infinite series in various fields like physics and engineering.
For geometric series, which include terms derived from multiplying a starting value by a constant ratio, we have a straightforward criterion for convergence. The series converges if the absolute value of the common ratio \(z\) is less than one, denoted as \(|z| < 1\). This ensures that as you add more terms, their impact on the sum becomes smaller and smaller, eventually stabilizing to a limit.
In the given exercise, the common ratio \(z = \frac{1}{3}\) meets this requirement, confirming that the series converges. Understanding convergence is key to practicing and applying infinite series in various fields like physics and engineering.
Complex Numbers
Complex numbers are fascinating mathematical constructs that extend our number system beyond real numbers. They take the form \(a + bi\), where \(a\) represents the real part and \(b\) the imaginary part, with \(i\) being the square root of \(-1\).
In the exercise, our initial term is \(a = 4i\), so we are dealing with a purely imaginary number, where the real part is zero. Complex numbers are not only abstract concepts; they play significant roles in fields such as electrical engineering and quantum physics. They help in solving equations where real number solutions do not exist and provide tools for modeling waveforms and oscillations.
Manipulating complex numbers is quite similar to algebra, with some added rules: remember \(i^2 = -1\). When calculating with complex numbers, like in our exercise where they form part of the series, it's crucial to properly apply these rules for accurate results. This understanding allows you to handle calculations involving imaginary units with confidence.
In the exercise, our initial term is \(a = 4i\), so we are dealing with a purely imaginary number, where the real part is zero. Complex numbers are not only abstract concepts; they play significant roles in fields such as electrical engineering and quantum physics. They help in solving equations where real number solutions do not exist and provide tools for modeling waveforms and oscillations.
Manipulating complex numbers is quite similar to algebra, with some added rules: remember \(i^2 = -1\). When calculating with complex numbers, like in our exercise where they form part of the series, it's crucial to properly apply these rules for accurate results. This understanding allows you to handle calculations involving imaginary units with confidence.
Sum of Series
Calculating the sum of an infinite series might seem daunting, but with geometric series, it becomes straightforward thanks to a specific formula. For a series with initial term \(a\) and a common ratio \(z\) that converges (where \(|z| < 1\)), the sum \(S\) can be determined using the formula:
- \(S = \frac{a}{1-z}\)
- Calculate the denominator: \(1 - z = 1 - \frac{1}{3} = \frac{2}{3}\)
- Substitute the values into the formula: \(S = \frac{4i}{\frac{2}{3}}\)
- Simplify: \(S = 4i \times \frac{3}{2} = 6i\)
Other exercises in this chapter
Problem 13
$$\begin{aligned}&S_{n}=\frac{1}{1+2 i}-\frac{1}{2+2 i}+\frac{1}{2+2 i}-\frac{1}{3+2 i}+\frac{1}{3+2 i}-\frac{1}{4+2 i}+\cdots+\frac{1}{n+2 i}-\frac{1}{n+1+2 i}
View solution Problem 16
$$\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+4\right)^{3}} d x=0 \quad(\text { The integrand is an odd function })$$
View solution Problem 17
We identify \(a=i / 2\) and \(z=i / 2 .\) since \(|z|=1 / 2
View solution Problem 18
$$\begin{aligned} f(z) &=\frac{1}{3(z+1)}+\frac{2}{3} \cdot \frac{1}{(z+1)-3}=\frac{1}{3(z+1)}+\frac{2}{3(z+1)} \cdot \frac{1}{1-\frac{3}{z+1}} \\ &=\frac{1}{3(
View solution