Find the radius of convergence and interval of convergence for ∑ n = 1 ∞ n x n 2 n

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Find the radius of convergence and interval of convergence for       ∑          n      =      1        ∞              n              x        n                    2      n

Harmony Horn · Accepted Answer

Here       a    n    =            n              x        n                    2      n            a          n      +      1        =  (  n  +  1  )  ⋅            x              n        +        1                    2              n        +        1              =  (  n  +  1  )                    x        n            ⋅      x                      2        n            ⋅      2      Now, by the ratio test,  L  =      lim          n      →      ∞                  a              n        +        1                    a      n        =      lim          n      →      ∞        =      lim          n      →      ∞        (  1  +      1    n    )  ⋅      x    2    =      lim          n      →      ∞        (  1  +      1    ∞    )      x    2      =      x    2  for cenvergence,  −  1  ≤  L  ≤  1    ⇒  −  1  ≤      x    2    ≤  1    ⇒  −  2  ≤  x  ≤  2So, Radius of convergence,   R  =            2      −      (      −      2      )        2    =  2now, interval of convergence  I  =  [  −  2  ,  2  ]

Find the radius of convergence and interval of convergence for sum_(n=1)^oo =(nx^n)/2^n

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