Note for MATH 153.01 at OSU
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Date Created: 02/06/15
Series Alternating Series Test Definition An alternating series is one who s terms flip back and forth between positive and negative Generally it contains 1quotn1 or lquotnl Test Two condition The numbers in the series have to be decreasing and the limit as n approaches infinity of En has to be zero If these are true then it converges If both are not satisfied then it diverges The Comparison Test Suppose that the summation of An and En have only positive terms If Bn converges and AnltBn then An diverges Basically you take a complex function with only positive integers and you use a simpler form with the dominate powers on the top and bottom of the function If the original is smaller than the simplified function and the simplified function converges then the complex function converges Estimating the error Sn a1aZa3an Sthe summation from nl to infinity of An RnSSn Rnerror Rnlt the integral from n to infinity of fxdx If you are asked to approximate the error in the first ten terms then you take the integral from 10 to infinity of the function Which will require you to replace infinity with T and take the limit as T approaches infinity of the integral from 10 to T of the function Then when you plug in the two integral bounds to the antiderivative you replace t with infinity and that s your answer Limit Comparison Test You take the original function and find a similar function with the main powers Similat to the comparison test You take the original function and divide it by the similar function When you get the answer you take the limit as n approaches infinity If this number is positive and finite then you know that the original equation behaves the same as the similar function Integral Test If we have the summation from nl to infinity of An and fx is given by Anfn and fx is a continuous increasing and positive function then the summation is only convergent if the integral from 1 to infinity of fxdx is convergent Which means that it goes to a finite number that is not infinity If the integreal converges then so does the series PSeries Use when you have the summation of one divided by n to any power Examples are lnquot2 lnquot3 lsqrtn or anything like that If pgtl then the series converges If pl or is lt1 it diverges Just simply state in the answer that is is convergent or divergent due to the Pseries test Properties of Series The summation of AnBn or AnBn the summation of An plus or minus the summation of En If the summation converges and has a constant then the constant can be brought outside of the summation Geometric series Example Sn aararquot2arquot3arquotnl AKA the summation from n1 to infinity of arquotn1 Basically something to the n power Like 2quotn or VzAn You solve these equations by using the formula alr a the first number in the series when you plug in what the first n is Usually nl R the number that is being raised to the n power in the equation This only works with a geometric series Limit theorem if the summation of An from n1 to infinity converges then the limit as n approaches infinity of An 0 So if you have a series and you take the limit of that series as n approaches infinity the answer has to equal zero If it does not equal zero then that means that the series is divergent If it equals zero then that means the the series converges It does not tell us what it converges to only that it converges Sequences Basically you take the sequence and then take the limit as n approaches infinity If you get a number then it is convergent You can use the squeeze theorem and sometimes l hospital s rule is necessary Use the squeeze theorem for most trig functions that are raised to a power and divided by something A sequence is increasing if a1lta2lta3lta4 and decreasing if a1gta2gta3gta4 When you have more complex functions you can tell if they are decreasing or increasing by taking the function and plugging in n Then you plug in n1 to the function If the original is less that the n1 function the sequence is increasing If it is the other way around then its decreasing You also have to check to make sure it is not oscillating which would mean that it has no monotonicity has to have an asymptote to converge 1032011 Absolute convergence if we have a summation from 1 to infinity of An we look at the absolute value of that series using only positive numbers obviously A series is absolutely convergent if the summation of the absolute value of An is convergent The original function is convergent ifthe absolute value of that function is convergent However ifa series is convergent that does not mean that it is absolutely convergent also When you put in the absolute values you can remove all 1quotn terms because they will not be relavent It seems most purposeful to use this when the alternating series test fails A series is llConditionally convergent if it is convergent but not absolutely convergent Ratio Test If the limit as n approaches infinity of the absolute value of An1An is less than 1 that means that the series is convergent and absolutely convergent If the answer is greater than one or equal to infinity then the series is divergent If the answer is equal to 1 then the test is inconclusive and you must try something else Root test Use this when you have a lot of variables that are raised to the n power You take the limit as n approaches infinity of the nth root of the absolute value of the series The nth root and the numbers power of n that the numbers are raised to cancel out leaving you with the absolute value of An If the answer is less than one then the series is convergent and absolutely convergent If the answer is greater than one or equal to infinity then the series is divergent If the answer is equal to 1 then the test is inconclusive and you must try something else
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