Answer Happy

Please do exercise number 11.
7. ΣΕ!(x – 5)*. k=0 8. Σ2*2%. k=0 k=0 9. Beginning with the geometric series which converges to 1- on (-1,1), find 1 power series which converge to and to arctanx on this same interval. 1+12 10. Let {ax} be a non-increasing sequence of non-negative numbers which con- verges to 0. Use Theorem 6.3.2 to show that the power series (-1)k+laxxx converges uniformly on [0, 1] and, hence, converges to a continuous function on this interval. 11. Use the preceding exercise and Example 6.4.11 to show that the alternating harmonic series 1 - 1/2+1/3 - ...-14+1/k + ... converges to In 2. Why do we need to use the previous exercise? Why isn't Example 6.4.11 enough? 12. Prove that if f(x) is the sum of a power series centered at a and with radius of convergence R, then f is infinitely differentiable on (a - R, a + R) - that is, its derivative of order m exists on this interval for all me N. 13. Suppose functions g and h are defined by 2.2k 9(α) = Σ h(α) = Σ. (2k + 1)!' (2k)! k=0 00 2.2k+1 k=0 Find the interval of convergence for each of these functions. 14. Prove that the functions in the previous exercise satisfy g' = h and h' = g. Thoorom 6.43