Please show work on how you get to the answer!

Answer 1 thru 7.

1. Determine the interval of convergence: 1 (a) _ 5n n=0 (b) S n5n n=1 2. Determine the interval of convergence: 18 1 + 2n (a) X" n=0 1 + 3n 1 + 2n (b ) 1=0 1 + 3n. 1 3. Find a power series for f (x) = (1 - x)3. Hint: use differentiation. 1 4. Find a power series for f (x) = Hint: Use 1 2+3x 1 - x = _ x" together with a little bit of algebra. 1=0 2 -3x 5. Find a power series for f (x) = 2x2 - 3x +1 Hint: First rewrite f(x) using partial fractions. 2n 6. Determine the interval of convergence of 1 + 3neh . 1=0 1 7. Write the function f (x) = 1 - 3x2 as a power series and determine the radius of convergence.Power Series & Representations of Functions as Power Serie Or A Whole New Way to Construct Functions of functions is about to be expanded. The main types of functions you use now are the about it), the logarithm and exponential functions, the root functions (square root, cube root and so on). polynomials (likely the first function functions you meth, such as f(x) = 3x3 + x - 9 and g(x) = 3(x-7/ -5(x-7) the the rational functions (quotients of polynomials). And of course, these functions can be combined in a Using infinite series, new types of functions can be created out of the blue. Power series are like poly- sorials gone wild while a polynomial adds up only a finite number of non-negative integer powers of the variable each multiplied by some coefficient, a power series is produced by adding up infinitely many such terms. So power series look like f (x ) = as tax tay tayt' t ...tend"+ ...= [aux". where the coefficients, so . $1, #2 . .. are some equal to 1, the power series would look like More generally, the variable I in the power series could be replaced by x - 2, or * + for x - c, where e is any constant. In that case the power series takes the form ( (x ) = as + a(x -c) + az(x -e)? + my(x -c)' + ... tau(x-ey + ...= [am(x-c)". Such a function is called a power series about orm given above. Plugging Numbers into Power Series ble. Unfortunately, computing the number is a little more work that computing the number a polynor Well, actually that's not too bad. We recognize that series as a geometric series, and we know the sum will of the series "-03'. But that series does not converge (the terms do not even approach (). Thus f(3) does not exist. In other words, 3 is not in the domain of that power series. Finding the Domain of a Power Series The first question we would like to be able to answer about a power series is: what is the domain? For ex ample, if f(x) = _ x", what is the domain of f? In other words, for which numbers x will the series converse? The Rock and Ratio Tests are usually the best tools to answer this question. For this particular example the ratio test tells us to compute lim = Lim Ixl = 1x the conclusion of the Ratio test then promises us that the series will con s less than 1, and it will diverge when the limit is greater than 1. Testing the Endpoints the limit is equal to 1, the test provides domain. We have learned nothing * = -1). Thus we know the domain of this power series is one Is: (-1,1). (-1.1). -1,1] or (-1, 1). To determine which one is correct, we test x = 1 and x = -1 in the ponce sew converges. For x - 1, we check S OF FUNCTIONS AS POWER SERIES 133 the series [ (-1)", and that also diverges Thus the domain of f(x) - [ " is the interval (-1,1)- CO The Compute the Dowarin Routine at example demonstrates the steps you would do to find the domain of a power series [ .(x - c)"- will produce an interval of a's centered at c inside of tered at c of some radius r. Outside that interval, the series will diverge. There will remain two points (the endpoints of the interval) where the convergence remains in doubt (x = 41 in our example, and x = c+r, x = c - in general). These two endpoints are then each tested in the series to see if they belong to the Power Series Vocabulary There is some specialized terminology associated with functions defined by power series. Since the domai of a power series always turns out be an interval, the phrase interval of convergence of the power series is used as a synonym for women of the power series, also the interval of convergence of a power series abou the radius of convergence of the power series. Notice that if you know the radius of conver series, you know almost everything about the domain. The only thing not clear is whether the er of the interval specified belong to the domain of the power series. For example, if we about o, and are told that the radius of convergence is 3, then we know the domain of the one of the intervals (-3,3). [-3,3). (-3,3] or [-3, 3). but we don't know which one. Algebra and Calculus With Power Series One of the nicest things about polynomials is that they are so easy to differentiate and integrate. These two them like polynomials' Differentiate and integrate term by terms. For example, if we need the derivative of ((x]- 1+s+8 +8+ +8+.-[r, the result is f(x) - 0+1+ 2r +38 + +mr"-14.. [ur- - [mr"-1. (The first term is 0, and so there is no harm starting the series with i = 1.) The polynomial. story is pretty much the same for integration. When it comes to integrating, treat a power series just like a The Effect of Calculus Operations convergence does not change. Warning ce might change. In other w power series is differentiated, the don have the same radius of convergence as the original power seri points might be lost or gained. CO Manipulating Known Power Series to Form New Power Series inciples is the buik series. For example, starting with the known power series nitegrating both sides yields, for |x]