MATH 118 Final Review Package

MATH 118 Final Review Package - SEQUENCES AND SERIES Taylor...

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S EQUENCES AND S ERIES Taylor and Maclaurin Polynomials. Example. Find the Maclaurin Polynomial for the function Find the first three derivatives of and evaluate them at : Substitute values in for the formula for the Maclaurin Polynomial: Example. Find the max possible error in using the Taylor Polynomial to approximate for . Find the sixth derivative of at where . . On this interval, (at ). Applying the formula for maximum error and substituting :
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Power Series. Example. Find the Taylor series about 5 for : Rearrange the expression to a form of : Substitute for the expression with and : Place restrictions on : Example. Find the sum of the power series Rearrange the expression in a different form: Recognize the sum as the power series for with as :
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Example. Find the sum of the power series . Find the radius of convergence: Let represent the sum. Multiply by and integrate to get rid of factor : Factor out and recognize the sum as the power series for with as : Differentiate to find :
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Infinite Sequences. An infinite sequence of numbers, indexed over the integers, can converge to a limit as n tends to infinity, or diverge. If a sequence is increasing or decreasing for all n, then it is said to be monotonic . A monotonic sequence that has an upper or lower bound is convergent. Properties of Limits of Sequences If r > 0, then If |r| < 1, then If , then If and is continuous at , then Example. Determine the limit for each of the following sequences. If a limit does not exist, state whether the sequence diverges to infinity, minus infinity, or neither. a) b) c) a) is continuous at so move the limit inside: b) Rearrange expression and apply l'Hôpital's rule: c) Divide by :
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Example. Let be defined by and . Find the limit. Set Bringing the limit in, and substituting L: Multiplying across by L: Solving the quadratic: Since ,
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Oscillating Sequences Example. Show that the sequence is convergent and find its limit. Calculate the first few terms to get an idea of the sequence: The first few terms oscillate, so use general case to prove the entire sequence oscillates: All terms of the sequence are positive, the denominator is positive. has the opposite sign of , proving the sequence is oscillating for all . Now, try and show approaches zero. Since all the terms of the sequence are positive, is always decreasing and has limit zero. Now that we have proven the sequence has a limit, we can calculate it.
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In order to find the root of a function, we can represent it as a sequence, and apply the method of successive approximation. Example. Consider the function . Apply the Successive approximation method to construct a sequence which can be used to estimate a root of the given function. Give the general recursive formula for the term of this sequence. If , give the value of .
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MATH 118 Final Review Package - SEQUENCES AND SERIES Taylor...

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