To determine Find whether the sequence is converge or diverge and find the limit of the convergent sequence. Explanation Given information: The n th formula for the sequence function is a n = sin 2 n 2 n . Theorem 4:Sandwich Theorem: Suppose that g ( x ) ≤ f ( x ) ≤ h ( x ) for all x in some open interval I containing c , except possibly at x = c itself. Suppose also that lim x → c g ( x ) = lim x → c h ( x ) = L . Then lim x → c f ( x ) = L . Convergent Definition: The sequence { a n } converges to the number L if for every positive number ε there corresponds an integer N such that | a n − L | < ε whenever n > N . If no such number L exists, we say that { a n } diverges. If { a n } converges to L , we write lim n → ∞ a n = L or simply a n = L , and call L the limit of the sequence. Divergent Definition: The sequence { a n } diverges to infinity if for every number M there is an integer N such that for all n larger than N , a n > M . If this condition holds, we write lim n → ∞ a n = ∞ or a n → ∞

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

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Author: Joel Hass, Christopher Heil

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 9.1, Problem 50E

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Find whether the sequence is converge or diverge and find the limit of the convergent sequence.

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Chapter 9.1, Problem 49E

Chapter 9.1, Problem 51E

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?

Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

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