Math 8 Quiz Due 07/23/14 01:20 PM

Math 8 Quiz Due 072314 0120 Pm

Write answers neatly on separate sheets of good-quality paper. Do not cramp results on one or two pages for the sake of saving paper! You must show your own work even if you work with others or if you take advantage of online resources. Sufficient explanation and proper step-wise reasoning are needed when you perform a test of convergence or divergence of a series. For Problems 1 and 2 use the following theorem: If {a_n} is convergent, then lim_n→∞ a_n = 0.

A sequence {a_n} is given recursively by a₁ = 12, and a_n+1 = 1/2 a_n.

• (a) Write down the first four terms of the sequence.
• (b) It is known that {a_n} is a convergent sequence. Find lim_n→∞ a_n.

A sequence {a_n} is given recursively by a₁ = 1, and a_n+1 = 1 + 1/a_n.

• (a) Write down the first four terms of the sequence.
• (b) It is known that {a_n} is a convergent sequence. Find lim_n→∞ a_n.
• (c) Use the results from (a) and (b) to estimate the value of 2.

3. Find a formula for the nth partial sum S_n of the series and use it to determine if the series converges or diverges. If the series converges, find its sum by evaluating lim_n→∞ S_n.

• (a) Series: ∑_n=1^∞ n / (n + 1)
• (b) Series: ∑_n=1^∞ 1 / n²
• (c) Series: ∑_n=1^∞ (ln(2) – ln(1 + 1/n))
• (d) Series: ∑_n=1^∞ 4 cos 3(5ⁿ)

4. Consider the series ∑_n=1^∞ n / (n!)

• (a) Use the ratio test to show that the series converges.
• (b) Find and express each partial sum S_n as a reduced fraction. Use the pattern to guess a formula for S_n.
• (c) Find the sum of the series.

5. Determine if the following series converge absolutely, converge conditionally, or diverge. Show the basis for your conclusion, such as applying an integral test or other convergence tests, ensuring the prerequisites are met.

• (a) ∑_n=1^∞ (ln n) / n
• (b) ∑_n=1^∞ (2 / n) sin n
• (c) ∑_n=1^∞ (ln n) / n²
• (d) ∑_n=2^∞ 1 / (n ln n)
• (e) ∑_n=1^∞ n^–n
• (f) ∑_n=1^∞ sin n / n
• (g) ∑_n=1^∞ n! / (3ⁿ)
• (h) ∑_n=1^∞ 1 / n^p
• (i) ∑_n=1^∞ 3 log n / n²
• (j) ∑_n=1^∞ arctan (1 / n)

6. Show that lim_n→∞ (2 / n)! = 0.

7. Solve the equation 2x³ + 5x = 0.

8. Use the series ∑_n=0^∞ xⁿ = 1 / (1 – x), |x| n=1^∞ a_n xⁿ. Determine the interval and radius of convergence.

9. Find the radius and interval of convergence of each power series:

• (a) ∑_n=1^∞ xⁿ
• (b) ∑_n=1^∞ nⁿ xⁿ

Paper For Above instruction

Introduction

The study of sequences and series is fundamental in calculus and mathematical analysis, serving as essential tools in understanding limits, convergence, and the behavior of infinite processes. This paper explores various problems related to sequences, series, convergence tests, power series, and limits, providing detailed solutions and theoretical insights into each topic.

Sequences and their Limits

We begin by analyzing recursive sequences and their limits, central to understanding infinite behavior. The sequences {a_n} given by a₁=12 with a recurrence relation a_n+1=1/2 a_n demonstrate geometric decay. The first four terms are computed as 12, 6, 3, 1.5, illustrating convergence towards zero. Applying the limit, we set L = 1/2 L, which yields L=0. Thus, lim_n→∞ a_n = 0, confirming the convergence of this sequence. Similarly, the sequence with a₁=1 and a_n+1 = 1 + 1/a_n converges to a fixed point satisfying L = 1 + 1/L. Solving L² - L -1=0 yields L=(1+√5)/2, approximately 1.618, which is the known golden ratio. This limit helps estimate its relation to the number 2, providing an approximation confidence in convergence behavior.

Series and Partial Sums

The evaluation of the sum of infinite series is crucial. For the series ∑ n/(n+1), telescoping provides the partial sum S_n = n - 1 + 1/(n+1), which converges as n approaches infinity to 1. The series ∑ 1/n² converges to π²/6, a famous result. The series with logarithmic terms, ∑ (ln 2 – ln(1+1/n)), telescopes to ln 2, ensuring convergence. The series involving cosine, ∑ 4 cos 3(5ⁿ), oscillates and requires more advanced analysis; it converges conditionally due to bounded oscillations.

Tests of Convergence

The ratio test applied to ∑ n / n! confirms absolute convergence, as the ratio of consecutive terms approaches zero. Partial sums computed manually reveal the pattern S_n = 1 – 1 / n!, leading to the conclusion that the sum is e – 1, using the exponential series expansion. For other series, tests such as the integral test, comparison, and alternating series test determine convergence type. For example, ∑ (ln n)/n diverges by comparison, while ∑ n^–n converges absolutely due to exponential decay.

Limits and Power Series

The limit lim_n→∞ (2/n)! is zero because factorial growth surpasses exponential growth, ensuring the limit's value is zero. Solving 2x³ + 5x = 0 involves factoring out x, which yields solutions x=0, and x=–2/√5, x=2/√5. For power series, the representation of 1 – ln(1 – x) can be derived from the Taylor expansion of ln(1 – x). Using the geometric series, the power series ∑_n=1^∞ xⁿ converges for |x| n=1^∞ nⁿ xⁿ converges based on the root test in a specific interval which can be numerically approximated.

Conclusion

This comprehensive analysis demonstrates the interconnectedness of sequences, series, convergence tests, and power series in calculus. Each problem's solution exemplifies critical techniques such as recursive analysis, telescoping sums, ratio and root tests, and power series expansion. These tools empower mathematicians and students to investigate the behavior of infinite processes rigorously, advancing understanding in mathematical analysis.

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