Calculus II - Power Series

4 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Power series

   If xxx is a variable, then a series of the form ∑k=0∞akxk=a0+∑k=1∞akxk=a0+a1x+a2x2...\sum_{k=0}^\infty a_kx^k=a_0+\sum_{k=1}^\infty a_kx^k=a_0+a_1x+a_2x^2...∑k=0∞​ak​xk=a0​+∑k=1∞​ak​xk=a0​+a1​x+a2​x2... where the coefficients aka_kak​ are constants, is called a power series in xxx or a power series centered at 000 . A series in the form ∑k=0∞ak(x−c)k=a0+∑k=1∞ak(x−c)k=a0+a1(x−c)+a2(x−c)2...\sum_{k=0}^\infty a_k(x-c)^k=a_0+\sum_{k=1}^\infty a_k(x-c)^k=a_0+a_1(x-c)+a_2(x-c)^2...∑k=0∞​ak​(x−c)k=a0​…

2. Convergence/Divergence of a power series

   If the power series ∑k=0∞akxk\sum_{k=0}^\infty a_kx^k∑k=0∞​ak​xk converges for a number x0≠0x_0 \neq 0x0​=0 , then it converges absolutely for all number xxx for which ∣x∣<∣x0∣|x|<|x_0|∣x∣<∣x0​∣ If the power series ∑k=0∞akxk\sum_{k=0}^\infty a_kx^k∑k=0∞​ak​xk diverges for some number x1x_1x1​ , then it diverges for all numbers xxx for which |x_1|"> ∣x∣>∣x1∣|x|>|x_1|∣x∣>∣x1​∣

3. Convergence types for a power series

   For a power series ∑k=0∞ak(x−c)k\sum_{k=0}^\infty a_k(x-c)^k∑k=0∞​ak​(x−c)k , exactly one of the following is true The series converges only if x=cx=cx=c The series converges absolutely for all xxx There is a positive number RRR for which the series converges absolutely for all xxx , ∣x−c∣<R|x-c|<R∣x−c∣<R , and diverges for all xxx , R"> ∣x−c∣>R|x-c|>R∣x−c∣>R . The behavior of the series at ∣x−c∣=R|x-c|=R∣x−c∣=R must be determined separately. RRR is defined as the radius of convergence. The set…

4. Properties of a power series

   Suppose ∑k=0∞akxk\sum_{k=0}^\infty a_kx^k∑k=0∞​ak​xk is a power series in xxx having a nonzero radius of convergence RRR Continuity property lim⁡x→x0(∑k=0∞akxk)=∑k=0∞(lim⁡x→x0akxk)=∑k=0∞akx0k\lim_{x\to x_0}(\sum_{k=0}^\infty a_kx^k)=\sum_{k=0}^\infty (\lim_{x\to x_0}a_kx^k)=\sum_{k=0}^\infty a_kx^k_0limx→x0​​(∑k=0∞​ak​xk)=∑k=0∞​(limx→x0​​ak​xk)=∑k=0∞​ak​x0k​ Differentiation property ddx(∑k=0∞akxk)=∑k=0∞(ddxakxk)=∑k=1∞kakxk−1\frac{d}{dx}(\sum_{k=0}^\infty a_kx^k)=\sum_{k=0}^\infty (\frac{d}{dx}a…