Problem 1 - Advanced Engineering Mathematics

Question

$$\frac{z}{1+z}=z\left[1-z+z^{2}-z^{3}+\cdots\right]=z-z^{2}+z^{3}-z^{4}+\cdots=\sum_{k=1}^{\infty}(-1)^{k+1} z^{k} ; \quad R=1$$

Step-by-Step Solution

Answer

The series sums to \(\frac{z}{1+z}\) for \(|z| < 1\).

Step 1: Understand the Given Series

We are given a series: \(\sum_{k=1}^{\infty}(-1)^{k+1}z^{k}\). This is an alternating series where the powers of \(z\) increase starting from \(z^1\).

Step 2: Identify the Series Type

Recognize the series as a geometric series. The series is of the form \(z - z^2 + z^3 - z^4 + \cdots\). This is a geometric series with the first term \(a = z\) and common ratio \(r = -z\).

Step 3: Find the Formula for the Sum of a Geometric Series

The sum \(S\) of an infinite geometric series with the first term \(a\) and common ratio \(r\) is given by \(S = \frac{a}{1-r}\) when \(|r| < 1\).

Step 4: Apply the Formula

For this series, we have \(a = z\) and \(r = -z\). Applying the formula, the sum is \(S = \frac{z}{1 - (-z)} = \frac{z}{1 + z}\).

Step 5: Validate the Radius of Convergence

The given radius of convergence \(R = 1\) is consistent with the condition \(|r| = |z| < 1\) for convergence, as \(r = -z\). This confirms that the series converges for \(|z| < 1\).

Step 6: Conclusion

The expression \(\frac{z}{1+z}\) is indeed equal to the sum of the given series \(\sum_{k=1}^{\infty}(-1)^{k+1} z^{k}\) for \(|z| < 1\).

Key Concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. In mathematical notation, it is expressed through series like

• \(a_1 - a_2 + a_3 - a_4 + \cdots\)
• or \((-1)^{k+1} a_k\) for each term \(a_k\)

In the context of the given series \(\sum_{k=1}^{\infty}(-1)^{k+1}z^{k}\), the series alternates in sign: positive, negative, positive, etc.
The series begins with the first positive term and each subsequent term's sign alternates. This is why you see the pattern of

• \(z - z^2 + z^3 - z^4 + \cdots\)

The use of alternating series can help in determining if a series converges, but measuring convergence relies on other methods too, like the Alternating Series Test. This test checks if the terms \(a_k\) decrease in absolute value to zero, supporting the convergence assurance for alternating series.

Radius of Convergence

The radius of convergence is a crucial concept when analyzing power series similar to the one given in the exercise. It defines the distance from the center of convergence within which the series will converge. Simply put, it is the value \(R\) that dictates where the values of our variable \(z\) can lie:

• Inside the radius (\(|z| < R\)), the series converges.
• On the boundary (\(|z| = R\)), convergence needs to be checked separately.
• Outside the radius (\(|z| > R\)), the series diverges.

In our exercise, the radius of convergence is given as \(R = 1\), meaning the series converges for any complex number \(z\) as long as \(|z| < 1\).
Understanding the radius of convergence is essential for determining the values for which an infinite series representation is valid, ensuring that manipulation and use of the series are mathematically sound.

Infinite Series

An infinite series is simply a sequence of terms added together that continues indefinitely. These can represent functions, calculations, or even certain constants under conditions of convergence. The series given in the exercise, \(\sum_{k=1}^{\infty}(-1)^{k+1}z^{k}\), illustrates an infinite series.
One common type of infinite series is the geometric series, recognized by a constant ratio between consecutive terms. Our series is indeed geometric with a common ratio of \(-z\).

• In the case of geometric series, it can sum to a finite value only if the absolute value of the common ratio, \(|r|\), is less than 1.

When dealing with infinite series, special attention must be paid to whether they converge or diverge. Convergence indicates that adding more terms will get the sum closer to a unique value. However, divergence suggests no limit is achievable from the addition of terms.
Infinite series find applications in approximations and deeper mathematical applications such as Fourier series, Taylor series, and other functions. Understanding their behavior enables predicting long-term results in many practical settings.