Problem 13 - Advanced Engineering Mathematics

Question

$$\begin{aligned}&S_{n}=\frac{1}{1+2 i}-\frac{1}{2+2 i}+\frac{1}{2+2 i}-\frac{1}{3+2 i}+\frac{1}{3+2 i}-\frac{1}{4+2 i}+\cdots+\frac{1}{n+2 i}-\frac{1}{n+1+2 i}=\frac{1}{1+2 i}-\frac{1}{n+1+2 i}\\\&\text { Thus, } \lim _{n \rightarrow \infty} S_{n}=\frac{1}{1+2 i}=\frac{1}{5}-\frac{2}{5} i \end{aligned}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{5} - \frac{2}{5} i \).

Step 1: Understand the Series Pattern

Identify the pattern from the given series: \(S_n = \frac{1}{1+2i} - \frac{1}{2+2i} + \frac{1}{2+2i} - \frac{1}{3+2i} + \frac{1}{3+2i} - \frac{1}{4+2i} + \cdots \). The series alternates in adding and subtracting fractions with complex denominators, resulting in a telescoping series.

Step 2: Simplify the Series

Notice that the series alternates in such a way that intermediate terms cancel each other: \( \frac{1}{2+2i} - \frac{1}{2+2i} = 0 \), \( \frac{1}{3+2i} - \frac{1}{3+2i} = 0 \), etc. The only terms that do not cancel out are \( \frac{1}{1+2i} \) (first term) and \( -\frac{1}{n+1+2i} \) (last term). Thus, \( S_n = \frac{1}{1+2i} - \frac{1}{n+1+2i} \).

Step 3: Evaluate the Limit

To find the limit as \( n \to \infty \), observe that the term \( \frac{1}{n+1+2i} \) approaches zero because the magnitude of its denominator increases without bound. Therefore, \( \lim_{n \to \infty} S_n = \frac{1}{1+2i} \).

Step 4: Convert to Rectangular Form

Convert \( \frac{1}{1+2i} \) into rectangular form. Multiply numerator and denominator by the conjugate of the denominator: \( \frac{1}{1+2i} \cdot \frac{1-2i}{1-2i} = \frac{1-2i}{1 + 4} = \frac{1-2i}{5} \). Hence, \( \frac{1}{1+2i} = \frac{1}{5} - \frac{2}{5}i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that comprise a real part and an imaginary part. The imaginary part is indicated by the symbol \( i \), representing \( \sqrt{-1} \). These numbers are expressed in the form \( a + bi \), where:

• \( a \): The real part, a standard real number.
• \( b \): The coefficient of the imaginary part.

For instance, in the complex number \( 3 + 4i \), 3 is the real part, and 4 is the imaginary part's coefficient.
Complex numbers are powerful in mathematics. They are essential for solving equations that have no real solutions, particularly quadratics with negative discriminants. These numbers are added, subtracted, multiplied, and divided in ways similar to algebraic variables. However, it's crucial during multiplication to include the property \( i^2 = -1 \).

Rectangular Form

The rectangular form of a complex number is simply its standard notation: \( a + bi \). This form offers a straightforward way of representing complex numbers, making it convenient to perform arithmetic operations.
Sometimes, converting complex fractions to rectangular form requires multiplying by the complex conjugate. The complex conjugate of \( a + bi \) is \( a - bi \). This process eliminates the imaginary number in the denominator, simplifying the expression.
Consider the transformation of \( \frac{1}{1+2i} \):

• Multiply by the conjugate \( 1 - 2i \).
• The result becomes \( \frac{1 - 2i}{1+4} = \frac{1 - 2i}{5} \).
• Thus, \( \frac{1}{1+2i} = \frac{1}{5} - \frac{2}{5}i \), showing the rectangular form.

Limits of Sequences

In mathematics, the limit of a sequence is a fundamental concept describing the behavior of sequences as they grow indefinitely. Specifically, a sequence \( a_n \) has a limit \( L \) if the terms approach \( L \) as \( n \) increases.

For example, consider the sequence \( S_n = \frac{1}{1+2i} - \frac{1}{n+1+2i} \). As \( n \) tends to infinity, observe the term \( \frac{1}{n+1+2i} \). Its magnitude decreases because the denominator grows larger and larger. Thus:

• \( \lim_{n \to \infty} \frac{1}{n+1+2i} = 0 \).
• A means that our sequence \( S_n \) converges to \( \frac{1}{1+2i} \).

Limits are crucial in calculus and analysis, serving as the backbone for defining continuity, derivatives, and integrals. They allow mathematicians to make rigorous conclusions about the behavior of functions and sequences over infinite expanses. Understanding limits helps in grasping more intricate mathematical concepts and solving complex problems.