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9. Determine the number of terms needed to approximate the sum of the convergent serie. an error of less than 0.001. a. n=0 00 b. Σ n=1 (-1)" 1 2"n! √e = -1)"+1 n² C. (-1)"+1 n=1 n4" = In 5|4

9. Determine the number of terms needed to approximate the sum of the convergent serie. an error of less than 0.001. a. n=0 00 b. Σ n=1 (-1)" 1 2"n! √e = -1)"+1 n² C. (-1)"+1 n=1 n4" = In 5|4

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with number 9 a to c. 

### Series Approximations and Convergence Tests #### 9. Approximating Series with an Error Threshold **Question 9:** Determine the number of terms needed to approximate the sum of the convergent series with an error of less than 0.001. - **(a)** \( \sum_{n=0}^{\infty} \frac{(-1)^n 2^n}{n!} = \frac{1}{\sqrt{e}} \) - **(b)** \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \) - **(c)** \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 4^n} = \ln \left( \frac{5}{4} \right) \) #### 10. Convergence and Divergence Testing **Question 10:** Use the root or ratio tests to determine the convergence or divergence of the series. If the test is inconclusive, use another test. State which test(s) you use. - **(a)** \( \sum_{n=1}^{\infty} \left( \frac{3}{4} \right)^n \) - **(b)** \( \sum_{n=0}^{\infty} \frac{3^n}{(n+1)^n} \) - **(c)** \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) - **(d)** \( \sum_{n=1}^{\infty} \frac{1}{4^n} \) - **(e)** \( \sum_{k=1}^{\infty} k \left( \frac{2}{3} \right)^k \) - **(f)** \( \sum_{n=1}^{\infty} \frac{\cos \frac{\pi n}{3}}{n!} \) - **(g)** \( \sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^2} \) - **(h)** \( \sum_{n=1}^{\infty} \frac{2^n n!}{5 \cdot 8 \cdot 11
Transcribed Image Text:### Series Approximations and Convergence Tests #### 9. Approximating Series with an Error Threshold **Question 9:** Determine the number of terms needed to approximate the sum of the convergent series with an error of less than 0.001. - **(a)** \( \sum_{n=0}^{\infty} \frac{(-1)^n 2^n}{n!} = \frac{1}{\sqrt{e}} \) - **(b)** \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \) - **(c)** \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 4^n} = \ln \left( \frac{5}{4} \right) \) #### 10. Convergence and Divergence Testing **Question 10:** Use the root or ratio tests to determine the convergence or divergence of the series. If the test is inconclusive, use another test. State which test(s) you use. - **(a)** \( \sum_{n=1}^{\infty} \left( \frac{3}{4} \right)^n \) - **(b)** \( \sum_{n=0}^{\infty} \frac{3^n}{(n+1)^n} \) - **(c)** \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) - **(d)** \( \sum_{n=1}^{\infty} \frac{1}{4^n} \) - **(e)** \( \sum_{k=1}^{\infty} k \left( \frac{2}{3} \right)^k \) - **(f)** \( \sum_{n=1}^{\infty} \frac{\cos \frac{\pi n}{3}}{n!} \) - **(g)** \( \sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^2} \) - **(h)** \( \sum_{n=1}^{\infty} \frac{2^n n!}{5 \cdot 8 \cdot 11
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