maths assignment

inconclusive. 7.2 Use the ratio test to determine whether the following series are convergent, divergent, or if your evaluation is (a) 5 + 22 23 + 24 + ... ( b) 3+ - 32 33 2 3+ 34 22 (d) 2+ - 23 24 4 (c) 32 + (e) 1 22 32 157 23 + 42 24+ ... (f) 3 -+ 32 33 417... 7.3 Determine the interval of convergence for the following power series. (a) 1+x+x2+x3+ ... (b) 1 x2 + 21 31 (c) 1 +x+- (d) x - 31 x4 (e) 1 2x + 3x2 4x + . (f) x - 7-4 Expand the following functions in a Maclaurin series (a) 1 + x (e) e- x2 (b ) (1 + x ) 2 (c) (1 4 x) 1/2 ax (d) In(1 - x) (g) coS X (h) (1+x)3 7-5 Show that for small values of XB the function In (1 - XB) - XB. 7.6 Show that for small values of e the function sine = 0. Homework exercises 7.7 Use your knowledge of power series to derive the following expressions for fundamental mathematical constants defined in terms of infinite series (a) e=) 1 1=0 n! (b) In2 - E-1)nil This infinite series representation of the mathematical constant e was used by Leonhard Euler (1707-1783) to provide an approximate value of e = 2.718281828459045235 to 18 decimal places. 7.8 Consider the power series [nx" 140 END-OF-CHAPTER PROBLEMS (a) Show that the interval of convergence of this series is (x]