5.5 Alternating Series

Alternating series are a fascinating part of calculus, with terms that switch between positive and negative. They're tricky because sometimes they converge even when the absolute values of their terms don't. This makes them unique and important to study.

Understanding alternating series helps us tackle real-world problems involving oscillating patterns. We'll learn how to test for convergence, estimate sums, and distinguish between absolute and conditional convergence. These skills are crucial for advanced math and physics applications.

Alternating Series

Alternating series test application

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• Alternating series test (Leibniz test) checks convergence of series with alternating signs
  • Series form $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ or $\sum_{n=1}^{\infty} (-1)^{n} b_n$ with $b_n > 0$ for all $n$
  • Sequence $\{b_n\}$ monotonically decreasing means $b_{n+1} \leq b_n$ for all $n$ (terms get smaller or stay the same)
  • $\lim_{n \to \infty} b_n = 0$ requires terms approach zero as $n$ increases (vanishing limit)
• Meeting all three conditions guarantees series convergence by alternating series test
  • Convergence assured but sum may not have closed form ($\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$ converges to $\ln(2)$)
  • Useful for determining convergence when ratio and root tests inconclusive ($\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$ converges by alternating series test)
• Oscillation of terms is a key characteristic of alternating series, with signs alternating between positive and negative

Sum estimation and error assessment

• Partial sum $S_n$ of alternating series sums first $n$ terms
  • $S_n = \sum_{k=1}^{n} (-1)^{k-1} b_k$ or $S_n = \sum_{k=1}^{n} (-1)^{k} b_k$ depending on series form
  • Approximates total sum $S$ with finite number of terms ($S_5 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5}$ for $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$)
• Error $|R_n|$ in approximating sum $S$ with $n$-th partial sum $S_n$ bounded by absolute value of next term $|a_{n+1}|$
  • $|R_n| < |a_{n+1}|$ provides upper limit on approximation error ($|R_5| < \frac{1}{6}$ for previous example)
  • More terms in partial sum means smaller error bound (taking $S_{10}$ reduces error to $|R_{10}| < \frac{1}{11}$)
• Actual sum $S$ lies between consecutive partial sums $S_n$ and $S_{n+1}$
  • $S_n < S < S_{n+1}$ when partial sums increase ($S_5 < S < S_6$ for $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$)
  • $S_{n+1} < S < S_n$ when partial sums decrease (happens for $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}$ instead)

Absolute vs conditional convergence

• Absolute convergence means series $\sum_{n=1}^{\infty} a_n$ and series of absolute values $\sum_{n=1}^{\infty} |a_n|$ both converge
  • Removing signs does not affect convergence ($\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2}$ and $\sum_{n=1}^{\infty} \frac{1}{n^2}$ both converge)
  • Absolutely convergent alternating series also conditionally converge (stronger condition)
• Conditional convergence means series $\sum_{n=1}^{\infty} a_n$ converges but $\sum_{n=1}^{\infty} |a_n|$ diverges
  • Signs essential for convergence ($\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$ converges but $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges)
  • Conditionally convergent alternating series not absolutely convergent (weaker condition)
• Testing convergence type for alternating series:
  1. Check convergence using alternating series test
  2. If converges, test absolute convergence of $\sum_{n=1}^{\infty} |a_n|$
  3. Absolute convergence if $\sum_{n=1}^{\infty} |a_n|$ converges, conditional if it diverges
• Absolute convergence allows rearrangements, conditional convergence sensitive to term order

Sequences and Series

• A sequence is an ordered list of numbers, while a series is the sum of terms in a sequence
• Convergence criteria for alternating series depend on properties of both the sequence and series sum
• The alternating series test provides specific conditions for series convergence based on sequence behavior