Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems A First Course in Complex Analysis · Problem 7.5.32
Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka
Mathematics — Complex Analysis · ratio-test-radius-of-convergence
Problem Prove that, if lim k → ∞ ∣ c k + 1 c k ∣ \lim_{k\to\infty} \left|\frac{c_{k+1}}{c_k}\right| lim k → ∞ c k c k + 1 ∑ k ≥ 0 c k ( z − z 0 ) k \sum_{k\geq 0} c_k(z-z_0)^k ∑ k ≥ 0 c k ( z − z 0 ) k R = { ∞ if lim k → ∞ ∣ c k + 1 c k ∣ = 0 , lim k → ∞ ∣ c k c k + 1 ∣ otherwise. R = \begin{cases} \infty & \text{if } \lim_{k\to\infty} \left|\frac{c_{k+1}}{c_k}\right| = 0, \\ \lim_{k\to\infty} \left|\frac{c_k}{c_{k+1}}\right| & \text{otherwise.} \end{cases} R = ⎩ ⎨ ⎧ ∞ lim k → ∞ c k + 1 c k if lim k → ∞ c k c k + 1 = 0 , otherwise. Browse the problem library