23-32 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 23. 3 − 4 + 16 – 64 + .. - - 3 25. 10 - 2 + 0.4 0.08 + ... 26. 2+0.5 + 0.125 + 0.03125 +... ∞ 27. Σ 12(0.73)"-14 n=1 00 29. Σ n=1 31. Σ (-3) -1 4" - pan n=1 6"-1 27 24. 4 + 3 + 2 + 1/76 + 28. Σ n=1 π 00 30. Σ 5 πη 200 32. Σ η 3n+1 n=0 (-2)" 6.22n-1 n=1 3" να asgrovans 21
23-32 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 23. 3 − 4 + 16 – 64 + .. - - 3 25. 10 - 2 + 0.4 0.08 + ... 26. 2+0.5 + 0.125 + 0.03125 +... ∞ 27. Σ 12(0.73)"-14 n=1 00 29. Σ n=1 31. Σ (-3) -1 4" - pan n=1 6"-1 27 24. 4 + 3 + 2 + 1/76 + 28. Σ n=1 π 00 30. Σ 5 πη 200 32. Σ η 3n+1 n=0 (-2)" 6.22n-1 n=1 3" να asgrovans 21
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
23,27,29
![**Determine Convergence or Divergence of Geometric Series**
Consider the geometric series given for each problem below. Determine whether the series is convergent or divergent. If it is convergent, find its sum.
**Problems:**
23. \( 3 - 4 + \frac{16}{3} - \frac{64}{9} + \ldots \)
24. \( 4 + 3 + \frac{9}{4} + \frac{27}{16} + \ldots \)
25. \( 10 - 2 + 0.4 - 0.08 + \ldots \)
26. \( 2 + 0.5 + 0.125 + 0.03125 + \ldots \)
27. \( \sum_{n=1}^{\infty} 12(0.73)^{n-1} \)
28. \( \sum_{n=1}^{\infty} \frac{5}{\pi^n} \)
29. \( \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^n} \)
30. \( \sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^n} \)
31. \( \sum_{n=1}^{\infty} \frac{e^{2n}}{6^{n-1}} \)
32. \( \sum_{n=1}^{\infty} \frac{6 \cdot 2^{2n-1}}{3^n} \)
**Instructions:**
1. Identify the common ratio \( r \) for each geometric series.
2. Determine if \( |r| < 1 \) for convergence.
3. If the series is convergent, use the formula for the sum of a convergent geometric series:
\[
S = \frac{a}{1 - r}
\]
where \( a \) is the first term of the series.
**Note:**
To solve these problems, an understanding of geometric series properties, convergence criteria, and summation formulas is essential. Each problem requires careful identification of the common ratio and first term to assess convergence and calculate sums if applicable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d8d0c97-e66d-424a-9c52-d4c7d8464ad8%2F0187a969-5fbf-43eb-8c01-d7a137e16933%2Fsmvo9t8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine Convergence or Divergence of Geometric Series**
Consider the geometric series given for each problem below. Determine whether the series is convergent or divergent. If it is convergent, find its sum.
**Problems:**
23. \( 3 - 4 + \frac{16}{3} - \frac{64}{9} + \ldots \)
24. \( 4 + 3 + \frac{9}{4} + \frac{27}{16} + \ldots \)
25. \( 10 - 2 + 0.4 - 0.08 + \ldots \)
26. \( 2 + 0.5 + 0.125 + 0.03125 + \ldots \)
27. \( \sum_{n=1}^{\infty} 12(0.73)^{n-1} \)
28. \( \sum_{n=1}^{\infty} \frac{5}{\pi^n} \)
29. \( \sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^n} \)
30. \( \sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^n} \)
31. \( \sum_{n=1}^{\infty} \frac{e^{2n}}{6^{n-1}} \)
32. \( \sum_{n=1}^{\infty} \frac{6 \cdot 2^{2n-1}}{3^n} \)
**Instructions:**
1. Identify the common ratio \( r \) for each geometric series.
2. Determine if \( |r| < 1 \) for convergence.
3. If the series is convergent, use the formula for the sum of a convergent geometric series:
\[
S = \frac{a}{1 - r}
\]
where \( a \) is the first term of the series.
**Note:**
To solve these problems, an understanding of geometric series properties, convergence criteria, and summation formulas is essential. Each problem requires careful identification of the common ratio and first term to assess convergence and calculate sums if applicable.
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