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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §10.2 Convergence Tests: Positive Series, #55 · Calculus · Series

For the series … show that the ratio test fails

For the series $\frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} + \dots$ show that the ratio test fails. The roots $(a_{n})^{1/ n}$ do approach a limit $L$ . Find $L$ from the even terms $a_{2 k} = 1/ 2^{k}$ . Does the series converge?

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §10.2 Convergence Tests: Positive Series, #55

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