Question

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The series \(\sum_{n=0}^{\infty} \frac{\sin (x+\pi n)}{x+\pi n}\) plays an important role in signal processing. Show that \(\sum_{n=0}^{\infty} \frac{\sin (x+\pi n)}{x+\pi n}\) converges whenever \(0

Short answer

The minimum N ensuring error < 10^-15 is determined by assessing decreasing terms until \\sum_term < \\epsilon. The series converges for 0 < x < \\pi.

Step-by-step solution

Understanding the Alternating Series Test

Recall that an alternating series \( \sum (-1)^n a_n \) converges if \( a_{n+1} \leq a_n \) for all \( n \) and \( \lim_{n \to \infty} a_n = 0 \). Here, the series is not strictly alternating in the usual format, but we can explore its convergence by considering the absolute value and oscillating signs of the sine function in \( \sum_{n=0}^{\infty} \frac{\sin(x + \pi n)}{x + \pi n} \).

Convergence for the Given Series

Note that \( \sin(x+\pi n) \) oscillates between \(-1\) and \(1\). For \(0 < x < \pi\), this behavior affects the terms \( \frac{\sin(x+\pi n)}{x+\pi n} \), making them diminish as \( n \) grows since the denominator grows without bounds. Thus, by comparison with a known convergent series like \( \sum \frac{1}{x+\pi n} \), we conclude that our series converges.

Using Remainder Estimate to Find N

We use the remainder estimate for convergent series to determine \( N \), wherein the error of the \( N \)th partial sum is less than a specified bound \( \epsilon \). For simplicity, assume an error \( \epsilon \) like \( 10^{-15} \). The remainder for the series can be estimated and compared with this error.

Calculate the Terms of the Series

Calculate the values of \( \frac{\sin(x+\pi n)}{x+\pi n} \) for reasonable values of \( n \) starting from \( n=0 \), ensuring each term diminishes in magnitude. Use these computations to visually or manually assess where the remainder term falls below our error \( \epsilon \).

Determine N and Approximate the Consecutive Sum

Based on the diminishing behavior of the sequence, continue until the cumulative difference between successive partial sums is less than \( \epsilon \), thus determining the minimum \( N \). Approximate the sum to ensure accuracy to within this error.

Use Sine Sum Formula for Extended Verification

Utilize the hint concerning the sum of angles: \( \sin(x + \pi n) = \sin(x) \cos(\pi n) + \cos(x) \sin(\pi n) \) where \( \cos(\pi n) = (-1)^n \) and \( \sin(\pi n) = 0 \). This simplifies to \((-1)^n \sin(x)\), reinforcing the alternating nature and rapid decay of the terms.

Key concepts

Signal Processing

Signal processing is a critical area in engineering that deals with the analysis, manipulation, and transformation of signals. Signals can be audio waves, electromagnetic waves, or any sequence that conveys information. The series \( \sum_{n=0}^{\infty} \frac{\sin(x+\pi n)}{x+\pi n} \) plays a unique role in this field.
To understand its relevance, consider that signal processing often involves analyzing waveforms and determining their different frequency components. By decomposing signals into series like the one given, engineers can identify the individual sine and cosine functions at play.
This series converges for \( 0 < x < \pi \), ensuring that the signal can be appropriately analyzed over that range. Such convergence is crucial in ensuring that the signal processing tools provide accurate and reliable information. Therefore, recognizing convergence enables professionals to implement more precise adjustments and modifications to the signals they work with.

Remainder Estimate

In mathematical series, the remainder estimate plays a vital role in determining how accurately the sum of the partial series approximates the total infinite series. This concept, particularly within alternating series, allows for an understanding of potential errors when truncating the series after a certain number of terms.
To find the minimum \( N \) for which the error bound holds, we utilize the remainder estimate. With alternating series, the absolute value of the (n+1)th term gives the upper bound for the error when you're working only up to the nth term. This means, if we want the error to be less than a specific amount, say \( 10^{-15} \), we need to ensure that the terms diminish rapidly enough by then.
By calculating several terms of the series \( \frac{\sin(x+\pi n)}{x+\pi n} \), we can visually and numerically identify where the error of the Nth partial sum becomes negligible, defining the smallest effective \( N \). If an enormous number of terms is needed for such precision, it indicates a slowly converging series, an important consideration in practical calculations.

Sine and Cosine Functions

Sine and cosine functions are fundamental trigonometric functions, regularly appearing in mathematics, physics, and engineering. They express relationships in periodic phenomena, such as waves.
The exercise explores \( \sin(x+\pi n) \), a variant of the sine sum formula \( \sin(a + b) \, = \, \sin a \cos b + \cos a \sin b \). This expression simplifies computations involving angles, especially in signal processing, where phase shifts are common.
For the series \( \sum_{n=0}^{\infty} \frac{\sin(x+\pi n)}{x+\pi n} \), \( \sin(x+\pi n) \) transforms into \((-1)^n \sin(x)\) because \( \cos(\pi n) = (-1)^n \) and \( \sin(\pi n) = 0 \). This property conveniently shows the alternating pattern of the series, vital for testing convergence using the alternating series test.
Understanding these functions and how they manipulate series can provide significant insight into many practical applications, from tuning musical instruments to decoding signals in telecommunications.