If the series has only polynomial components and logarithms, use limit comparison test. If the series is of the form ∑( ), use root test. Otherwise, try the series tests in this order: Step 5. - Ratio Test -> Comparison Test -> Limit Comparison Test -> Root Test Choosing a Series to Compare to: The series we compare to should have the same convergence properties as the series we are examining. This requires an understanding of guessing if a series converges or not. Essentially we want to examine the series as n goes to infinity. This means: Usually we would compare to the series which we “discovered” our series behaveslike. We can ignore: 1) Logarithms that the series is being multiplied or divided by 2) Constants that are being added or subtracted 3) Terms that are being added or subtracted which aren’t the highest degree term∑( )So this series behaves like: ∑∑And therefore diverges 1 2 2 3 Ex:
Approximating Error on a Partial Series: The series that we have been looking at are all infinite series. However, we will sometimes approximate an infinite series with one which only has a finite number of terms. Like so: ∑ ∑ ∑The series on the far right, ∑, is our error, E. There are two ways we approximate it: Alternating Series: If the series is an alternating series, we simply say that the error is less than the next term in the series: Integral Test: If the series was not an alternating series, but is a series that the integral test can apply to, then: ∫
Power Series and Taylor Series What are Power Series and Taylor Series? Taylor’s FormulaA power series is a series which has a variable in it, whose power increases with each term. ∑ ( )is a general power series. Power series may converge or diverge at different values of x. Each series has a radius of convergence, R, and an interval of convergence, where x is betweena-R and a+R. A Taylor Series is a specific type of power series. Any differentiable function can generate a taylor series, through use of Taylor’s Formula:f(x) generates a series, centered at some value a: ∑( )( )( ). This can turn any function into a polynomial. We can split up this series like so: ∑( )( )( )∑( )( )( )∑( )( )( )Then the first part, ∑( )( )( ), we call Tm.The second part, ∑( )( )( )Operations on Power Series Addition and Subtraction ∑( )∑( )∑( )( )Essentially they are added and subtracted very simply. The trick is that the (x-a) term has to be common between both series, and the interval of convergence of the new series only covers the x values which cause both series to converge.Multiplication and Division ∑( )∑( )∑( (∑) ( ))The radius of convergence of the new series is the smaller of the two radii of convergences. Integrals and Derivatives ∫∑( ) ∑∫ ( ) ∑∫ ( )Radius of Convergence For a series, ∑ , the radius of convergence of U, RU,is . If, for instance,then √
Converting Between Functions and Power Series ( )∑( )( ) ( ) ( ) ∑( )Step 1)Examine the list to the left and determine which series/function looks most like the one we’re given in the question.