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Find values of x for series converge?  `sum_(n=1)^oo 1/(2n-1)((x+2)/(x-1))^n`

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yapayapa | (Level 2) Honors

Posted August 31, 2013 at 2:41 PM via web

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Find values of x for series converge?

 `sum_(n=1)^oo 1/(2n-1)((x+2)/(x-1))^n`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted August 31, 2013 at 3:06 PM (Answer #1)

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The series `sum_1^oo 1/(2n - 1)((x+2)/(x-1))^n` converges when `lim_(n->oo)(|a_n|)^(1/n) < 1` .

For the series given `lim_(n->oo)(|a_n|)^(1/n)`

= `lim_(n->oo)(1/(2n - 1))^(1/n)((x+2)/(x-1))^(n/n)`

= `lim_(n->oo)(1/(2n - 1))^(1/n)((x+2)/(x-1))`

As `n->oo` , `1/n-> 0` and `(1/(2n - 1))^(1/n) -> 1` .

For the given condition to be satisfied, `(x + 2)/(x - 1) < 1` .

x + 2 is always greater than x - 1. For `(x + 2)/(x - 1) < 1` , either x + 2 = 0 or x + 2 > 0 while x - 1 < 0

x + 2 `>=` 0, x -1 < 0

=> -2 `<=` x < 1

The given series converges when `-2 <= x < 1`

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