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Determine if the alternating series converges: `sum_(n=2)^oo (-1)^(n)sin(1/n)`
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High School Teacher
By Leibniz criterion the series `sum_(n=1)^oo (-1)^n a_n` if
`a_1geq a_2geq a_3geq cdots` (1)
Let's now apply Leibniz criterion to our problem. In our case `a_n=sin(1/n)`.
Since `pi/2geq 1/n geq0` for `n in NN` and sinus is monotonically decreasing function over `[pi/2,0]` ` ` we have condition (1).
Let's now check limit
So both conditions required by Leibniz criterion are met and thus the series converges.
Posted by tiburtius on April 25, 2013 at 3:50 PM (Answer #2)
Let us define the series as
(i)ignoring the sign
`a_(n+1)<a_n` for all n
Therefore series `sum_(n=1)^oo(-1)^nsin(1/n)` will converge.
Posted by pramodpandey on April 25, 2013 at 3:51 PM (Answer #3)
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