Question

Test each of the following series for convergence. $$\sum \frac{1}{(1+i)^{n}}$$

Short answer

The series converges because the common ratio \(\frac{1}{\sqrt{2}}\) is less than 1.

Step-by-step solution

Identify the series

The series given is \(\sum \frac{1}{(1+i)^{n}}\). Note that \(1+i\) is a complex number.

Determine the modulus of the base of the exponent

The modulus of \(1+i\) is calculated as follows: \(|1+i| = \sqrt{1^{2} + 1^{2}} = \sqrt{2}\).

Rewrite the general term using the modulus

Rewrite the term \(\frac{1}{(1+i)^n}\) as \(\frac{1}{(\sqrt{2})^n}\) because the magnitude of \((1+i)^n\) is \(\sqrt{2}^n\). This simplifies to \(\left(\frac{1}{\sqrt{2}}\right)^n\).

Recognize the series type

The series is now a geometric series of the form \(\sum \left(\frac{1}{\sqrt{2}}\right)^n\).

Determine the common ratio

The common ratio \(r\) of the geometric series is \(\frac{1}{\sqrt{2}}\). Since \(\left| \frac{1}{\sqrt{2}} \right| < 1\), we can conclude that the series converges.

Key concepts

Complex Numbers

A complex number is a number that has both a real part and an imaginary part. It is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). Complex numbers extend the idea of one-dimensional number lines to two-dimensional complex planes.

For example, in the exercise, \(1+i\) is a complex number, where \(1\) is the real part and \(i\) is the imaginary part. Complex numbers are essential in various fields, including engineering, physics, and mathematics.

• Real Part (Re): The component without \(i\). In \(1+i\), this is \(1\).
• Imaginary Part (Im): The component with \(i\). In \(1+i\), this is \(1 \times i\).

Understanding complex numbers lays the groundwork for more advanced concepts, including the modulus and operations on complex numbers.

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (\(r\)). The general form of a geometric series can be written as:
\(\text{\textbackslash}\textbackslashsum a r^n\),
where \(a\) is the first term, and \(r\) is the common ratio.

From the exercise, after transformation, the series \(\text{\textbackslash}\textbackslashsum \frac{1}{(\textbackslash}\textbackslashsqrt{2})^n\) is recognized as a geometric series. Here, \(a\) is \(1\) and \(r\) is \( \text{\textbackslash}\textbackslashfrac{1}{\text{\textbackslash}\textbackslashsqrt{2}}\). Geometric series have interesting properties and simple tests for convergence. If \(|r| < 1\), the series converges.

Modulus of a Complex Number

The modulus (or magnitude) of a complex number \(z = a + bi\) is a measure of its size. It is denoted as \(|z|\) and is defined as:
\(|a + bi| = \text{\textbackslash}\textbackslashsqrt{a^2 + b^2}\)
This gives the distance between the point representing the complex number in the complex plane and the origin.

In the exercise, the modulus of \(1 + i\) is calculated as:
\(|1 + i| = \text{\textbackslash}\textbackslashsqrt{1^2 + 1^2} = \text{\textbackslash}\textbackslashsqrt{2}\).

The modulus helps in converting a complex number into polar form and in solving various problems involving complex numbers.

Common Ratio

The common ratio (\(r\)) in a geometric series is the factor by which we multiply each term to get the next term. It plays a critical role in determining whether the series converges or diverges. The general form has \(r\) such that:
\( r = \frac{\text{Term}_{n}}{\text{Term}_{n-1}}\)

In the exercise, the series \(\text{\textbackslash}\textbackslashsum \frac{1}{(1+i)^n}\) is transformed into a geometric series by recognizing that the modulus \(\text{\textbackslash}\textbackslashsqrt{2}\) simplifies the term to \(\text{\textbackslash}\textbackslashleft( \frac{1}{\text{\textbackslash}\textbackslashsqrt{2}} \text{\textbackslash}\textbackslashright)^n\). Here, the common ratio is \(r = \frac{1}{\text{\textbackslash}\textbackslashsqrt{2}}\). Because \(|r| < 1\), we conclude that the geometric series converges.

• If \(|r| < 1\), the series converges.
• If \(|r| \text{\textgreater} 1\), the series diverges.
• If \(|r| = 1\), it depends on the initial term and the form of the series.

Recognizing the common ratio is crucial for determining the behavior of geometric series.