Approximating Infinite Series Sums A Step-by-Step Guide

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In mathematics, especially in calculus and analysis, understanding the convergence and sum of infinite series is a crucial concept. While some series have a straightforward formula for their sum, many others require approximation techniques. This article delves into approximating the sum of an alternating series using the first few terms, specifically focusing on the example of the series: $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{4^n} $. We will explore the theory behind alternating series, the Alternating Series Test, and how to apply the Alternating Series Estimation Theorem to find accurate approximations. Our main goal is to provide a comprehensive guide that not only answers the given problem but also enhances the understanding of series approximations in general.

The Essence of Infinite Series

An infinite series is the sum of an infinite number of terms. These series can either converge to a finite value or diverge, meaning their sum grows without bound. The series we are considering, $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{4^n} $, is an example of an alternating series. Alternating series are characterized by terms that alternate in sign, often due to a (−1)n(-1)^n or (−1)n+1(-1)^{n+1} factor. These series have unique properties that make them easier to approximate than other types of series.

Alternating Series Test

Before approximating the sum, it's essential to ensure that the series converges. The Alternating Series Test provides a straightforward criterion for convergence. An alternating series of the form $ \sum_{n=1}^{\infty} (-1)^{n+1} b_n $ or $ \sum_{n=1}^{\infty} (-1)^{n} b_n $ converges if the following two conditions are met:

  1. bn>0b_n > 0 for all nn
  2. The sequence {bnb_n} is decreasing, i.e., bn+1≤bnb_{n+1} \leq b_n for all nn
  3. $ \lim_{n \to \infty} b_n = 0 $

For our series, $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{4^n} $, we have $ b_n = \frac{n}{4^n} $. Let's verify these conditions:

  1. Clearly, $ \frac{n}{4^n} > 0 $ for all $ n \geq 1 $.

  2. To show that the sequence is decreasing, we can consider the function $ f(x) = \frac{x}{4^x} $ and examine its derivative.

    f′(x)=4x−ximes4xln⁡(4)(4x)2=1−xln⁡(4)4x f'(x) = \frac{4^x - x imes 4^x \ln(4)}{(4^x)^2} = \frac{1 - x \ln(4)}{4^x}

    Since $ \ln(4) \approx 1.386 $, $ 1 - x \ln(4) < 0 $ for $ x > \frac{1}{\ln(4)} \approx 0.721 $. Thus, $ f'(x) < 0 $ for $ x > 0.721 $, which means the function is decreasing for $ x > 1 $. Therefore, the sequence $ b_n = \frac{n}{4^n} $ is decreasing for $ n \geq 1 $.

  3. To show that the limit is zero, we consider $ \lim_{n \to \infty} \frac{n}{4^n} $. This is an indeterminate form of type $ \frac{\infty}{\infty} $, so we can apply L'Hôpital's Rule:

    lim⁡n→∞n4n=lim⁡n→∞14nln⁡(4)=0 \lim_{n \to \infty} \frac{n}{4^n} = \lim_{n \to \infty} \frac{1}{4^n \ln(4)} = 0

Since all three conditions are satisfied, the Alternating Series Test confirms that our series converges.

Alternating Series Estimation Theorem

The Alternating Series Estimation Theorem (ASET) provides a method for estimating the error when approximating the sum of a convergent alternating series by using a partial sum. The theorem states that if $ s = \sum_{n=1}^{\infty} (-1)^{n+1} b_n $ is the sum of a convergent alternating series satisfying the conditions of the Alternating Series Test, then the error ∣Rn∣=∣s−sn∣|R_n| = |s - s_n| (where sns_n is the nth partial sum) is no greater than the absolute value of the (n+1)(n+1)th term, i.e., ∣Rn∣≤bn+1|R_n| \leq b_{n+1}.

In simpler terms, if we add up the first nn terms of an alternating series, the error in our approximation is no larger than the absolute value of the next term we would have added.

Approximating the Sum Using the First Six Terms

To approximate the sum of the series $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{4^n} $ using the first six terms, we first calculate the partial sum $ s_6 $:

s6=∑n=16(−1)n+1n4n=14−242+343−444+545−646 s_6 = \sum_{n=1}^{6} \frac{(-1)^{n+1} n}{4^n} = \frac{1}{4} - \frac{2}{4^2} + \frac{3}{4^3} - \frac{4}{4^4} + \frac{5}{4^5} - \frac{6}{4^6}

Calculating each term:

  • $ \frac{1}{4} = 0.25 $
  • $ -\frac{2}{4^2} = -\frac{2}{16} = -0.125 $
  • $ \frac{3}{4^3} = \frac{3}{64} \approx 0.046875 $
  • $ -\frac{4}{4^4} = -\frac{4}{256} = -0.015625 $
  • $ \frac{5}{4^5} = \frac{5}{1024} \approx 0.0048828 $
  • $ -\frac{6}{4^6} = -\frac{6}{4096} \approx -0.0014648 $

Adding these terms together:

s6≈0.25−0.125+0.046875−0.015625+0.0048828−0.0014648≈0.159668 s_6 \approx 0.25 - 0.125 + 0.046875 - 0.015625 + 0.0048828 - 0.0014648 \approx 0.159668

Rounding to four decimal places, $ s_6 \approx 0.1597 $.

Error Estimation

According to the Alternating Series Estimation Theorem, the error ∣R6∣|R_6| is bounded by the absolute value of the 7th term, $ b_7 $:

∣R6∣≤b7=747=716384≈0.0004272 |R_6| \leq b_7 = \frac{7}{4^7} = \frac{7}{16384} \approx 0.0004272

This means our approximation $ s_6 $ is within 0.0004272 of the true sum SS. Therefore, we can write the inequality:

s6−b7≤S≤s6+b7 s_6 - b_7 \leq S \leq s_6 + b_7

Substituting the values:

0.159668−0.0004272≤S≤0.159668+0.0004272 0.159668 - 0.0004272 \leq S \leq 0.159668 + 0.0004272

0.1592408≤S≤0.1600952 0.1592408 \leq S \leq 0.1600952

Rounding to four decimal places:

0.1592≤S≤0.1601 0.1592 \leq S \leq 0.1601

Conclusion

By applying the Alternating Series Test and the Alternating Series Estimation Theorem, we successfully approximated the sum of the given infinite series $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{4^n} $ using the first six terms. We found that the sum SS lies within the interval $ 0.1592 \leq S \leq 0.1601 $. This approach highlights the power of these theorems in providing accurate estimations for convergent alternating series. Understanding these concepts is crucial for various applications in mathematics, physics, and engineering, where infinite series frequently arise. Through careful application of these principles, we can confidently approximate the sums of complex series and gain valuable insights into their behavior.

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  • Alternating Series: A core concept, used extensively throughout the article.
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