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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