T3.3 Differentiation of Power Series² This exercise is optional. Prove that Σkak k=1 = X (1 - x)²¹ |x| < 1. T3.3 Differe
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T3.3 Differentiation of Power Series² This exercise is optional. Prove that Σkak k=1 = X (1 - x)²¹ |x| < 1.
T3.3 Differentiation of Power Series We first check the radius of convergence of Σzk+1 k=0 lim 1 = 1 k→∞ as well as ∞ =rk. We have k=1 = 1 lim = 1. k→∞ 1 Therefore, we may invoke the Theorem of Cauchy-Hadamard or of D'Alembert to obtain r = 1 as the radius of convergence. Moreover, æk is a differentable function for x € Σkak = [(k+ 1)ak+¹ = xΣ(k+1)æk k=1 k=0 k=0 d d 2. · · · ( 1 +² +) @ + - ± (2 ++) ΙΣ =I. dx dx k=0 \k=0 k=1 (-1, 1) and we can interchange differentiation and summation for each x € (-1, 1): d d d 1 = - ²-² (2²¹) · (~ . Σ ^ ) - ² (²) ²² =x. dx dx dx (1-x)²* k=1 k=0 =X For (*) we used that interchangeability of differentiation and summation.