POWER SERIES DALEMBERTS RATIO TEST Learning Outcomes Be

POWER SERIES & D’ALEMBERT’S RATIO TEST Learning Outcomes: • Be able to find the sum to infinity of a Power series • Be able to test for absolute convergence using D’Alembert’s ratio test

• D’Alembert’s ratio test for absolute convergence (if signs of terms alternate) Øconvergence: Ødivergence: Øindeterminate:

POWER SERIES • Power Series eg

POWER SERIES & D’ALEMBERT’S RATIO TEST • Power Series • D’Alembert’s ratio test for absolute convergence of a Power Series Øconvergence: Ødivergence: Øindeterminate:

Example: Find the sum to infinity of the series Page 86 and find the values of x for which the sum is valid. The coefficients form an arithmetic sequence: we can use the difference d = 2 We will subtract x. Sn from Sn 0 Complicated method – for explanation 0

Example: Find the sum to infinity of the series Page 86 and find the values of x for which the sum is valid. The coefficients form an arithmetic sequence: we can use the difference d = 2 We will subtract x. S from S More sensible method

Example: Find the sum to infinity of the series Page 87 and find the values of x for which the sum is valid. The coefficients form a linear recurrence relation We will subtract x. S from S RHS is geometric r = 2 x

Example: Find the sum to infinity of the series Page 86 and find the values of x for which the sum is valid. The coefficients form the Fibonacci sequence We will subtract x. S from S The Golden ratio D’Alembert: S converges when Page 87 Ex 2