Review of Series Math 1b May 13 2005 1 Tests for Convergence 2 Strategies for Convergence Tests Take all lists with a grain of salt P 1 1 If the series is of the form np it is a p series which we know to be convergent if p 1 and divergent if p 1 3 Example Determine the convergence or divergence of the following series i ii iii X 1 1 2 n 1 n X 1 3 2 n 1 n X 1 n 1 n iv X 2 n0 001 n 1 4 P P n n 1 ar or ar it is a geometric sereis 2 If the series has the form which converges if r 1 and diverges if r 1 Some preliminary algebraic manipulation by be required to bring the series into this form 5 Example Determine the convergence or divergence of the following series X 1 i n n 1 3 ii X 3n8 n 1 n 1 iii X e n n 1 iv X 1 n 0 99999 n n 1 6 3 If the series has a form that is similar to a p series or geometric series then one of the comparison tests should be considered In particular if an is a rational function or algebraic function of n involving roots of polynomials then series should be compared with a p series The value of p should be chosen by keeping only the highest powers of the numerator and denominator The comparison tests apply only to series P with positive terms but if an has some negative terms then we can P apply the Comparison Test to an and test for absolute convergence 7 Example Determine the convergence or divergence of the following series X n2 1 i 2 n 1 n n X n2 1 ii 2 n n n 1 iii X n 4 2 n 1 n n n X q n2 1 iv 3 2 n 1 n 2n 5 8 4 If you can see at a glance that limn an 6 0 then the Test for Divergence should be used 5 If the series is of the form 1 nbn or 1 n 1bn then the Alternating Series Test is an obvious possibility P P 9 6 Series that involve factorials or other products including nth powers are often conveniently tested using the Ratio Rest Bear in mind that an 1 an 1 as n for all p series and therefore all rational or algebraic functions of n Thus the Ratio Test should not be used for such series 10 X n 3n Example Test the series for convergence n 1 1 8n 11 7 If an f n where 1 f x dx is easily evaluated then the Integral Test is effective assuming the hypotheses of this test are satisfied R 12 Example Determine the convergence of X 3 k2e k k 1 13 Advanced Strategies 14 X cos n 2 Example Test the series for convergence 2 n 1 n 4n 15 Crude Estimates sin x 1 cos x 1 arctan x 2 16 Logarithms When dealing with logarithms remember that they grow more slowly than any power function ln x 0 x xp lim x is eventually less than one as long as p 0 This means for all p ln p x In other words log x xp for x big enough 17 Example Determine the convergence of X k ln k 3 k 1 k 1 18 Exponentials Exponentials grow faster than any power function Meaning np 0 lim n en for all p 0 why So as long as n is big enough we have np 1 n 1 e p en n 19 Example Determine the convergence of X 3 k5e k k 1 20 Using Taylor Series to decide convergence 21 Example Test the series for convergence i X sin 1 n n 1 ii X n tan 1 n n 1 iii X tan 1 n n 1 n 22 Theory of Convergence 23 Midterm 2 Problem 2 Suppose n 1 an 0 6 and an 0 for all n Let sn a1 a2 an Which of the following statements must be true which must be false and for which is it impossible to decide Explain your reasoning P 24 a lim sn 0 n 25 b an 1 an for all n 26 a c limn n 1 an 1 27 a d limn n 1 an 1 28 e P n 1 sn converges 29 f P 2 n 1 an converges 30 g P n 1 ln an converges 31 h P n 1 ln 1 an converges 32 Power Series 33 Recognizing a series as the evaluation of a power series Suppose the series x 0 X x 2 nbn converges at x 4 and diverges at n 1 What is the interval of convergence 34 Continued Determine whether the following series converge i X 1 n2nbn n 1 ii X bn n 1 iii X 1 n 1 bn 35 Computing Taylor Series 36 Famous Series X 1 1 un 2 3 u e 1 u u u 2 3 n 0 n X 1 5 1 nx2n 1 1 3 sin u u u u 3 5 n 0 2n 1 X 1 2 1 5 1 nx2n cos u 1 u u 2 4 2n n 0 X 1 1 u u2 u3 un 1 u n 0 37 Example a Find a power series representation of f x e x b Use it to find a series converging to Z 0 2 2 x e dx 38 Example Let f x x3 arctan x2 Find f 100 0 39 Rates of Convergence or the dreaded Remainder Estimate 40 Shannon and Sayid are castaways on a tropical island To pass the time Shannon has baked a coconut cream pie She tells Sayid she will give him a piece if he can compute ln 2 and guarantee his error is no more than 10 10 41 Sayid has no calculator it got destroyed in the plane crash but is confident because he remembers his Math 1b very well He finds the power series for ln 1 x and plugs in x 1 Find the series Sayid claims converges to ln 2 42 Wait says Shannon Are you sure that series converges Find the radius of convergence of the power series in question 43 Okay says Sayid maybe we don t know for sure But I know the first few digits by heart and the series seems to converge to the right thing Assume Sayid adds terms together at the rate of one term every second He got an A in Math 1b How long before Sayid can guarantee he is within 10 10 of ln 2 44 I think I ll go see if Sawyer wants a piece says Shannon Hold on says Sayid I have another idea He realized that 9 3 ln 2 2 ln ln 2 8 Find Sayid s second series that converges to ln 2 45 Now how long before Sayid is eating …