Question
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Answer :
Word Count : 292
(i) We are asked to test the convergence of the series [ \sum_{n=1}^{\infty} n x^{,n-1}, \quad x > 0. ] We can use the ratio test. Let (a_n = n x^{,n-1}). Consider [ \frac{a_{n+1}}{a_n} = \frac{(n+1)x^n}{n x^{,n-1}} = \frac{n+1}{n} \cdot x = \left(1 + \frac{1}{n}\right)x. ] Take the limit as (n \to \infty): [ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = x. ] * If (x < 1), the ratio (< 1), series converges. * ________ ______ _______ ________ _________ ________ ______ ____ _____ ___.
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(i) We are asked to test the convergence of the series [ \sum_{n=1}^{\infty} n x^{,n-1}, \quad x > 0. ] We can use the ratio test. Let (a_n = n x^{,n-1}). Consider [ \frac{a_{n+1}}{a_n} = \frac{(n+1)x^n}{n x^{,n-1}} = \frac{n+1}{n} \cdot x = \left(1 + \frac{1}{n}\right)x. ] Take the limit as (n \to \infty): [ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = x. ] * If (x < 1), the ratio (< 1), series converges. * ________ ______ _______ ________ _________ ________ ______ ____ _____ ___.
__________ _________ ____ ________ _______ ________ _____.
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