how do you find the radius of convergence of the power series `(-1)^n ((x-2)^n/(n2^n))`    

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Write the nth term as `C_n (x-2)^n`

Using the ratio test, the series converges is

`lim_(n-> oo) |(C_(n+1)(x-2)^(n+1))/(C_n(x-2)^n)| < 1`

Where the radius of convergence is `lim_(n-> oo) |C_n/C_(n+1)|`

Now

`lim_(n-> oo) | (C_(n+1)(x-2)^(n+1))/(C_n(x-2)^n)| = lim_(n-> oo) |((-1)^(n+1)((x-2)^(n+1))n2^n)/((-1)^n((x-2)^n)(n+1)2^(n+1))|`

` `

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

This is less than 1 if

`| (x-2)/2| < 1` 

ie if `|x-2| < 2`

`therefore`  the radius of convergence is 2

 

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