Verified answer

By the Ratio Test,

r = lim(n→∞) |[x^(n+1)/((n+1)² 2^(n+1))] / [xⁿ/(n² 2ⁿ)]|

..= (1/2) |x| * lim(n→∞) n²/(n+1)²

..= (1/2) |x| * 1

..= (1/2) |x|.

Hence, the series converges for (1/2) |x| < 1 <==> |x| < 2 and diverges for |x| > 2.

Checking the endpoints:

x = 2 ==> Σ(n = 1 to ∞) 1/n², convergent p-series.

x = -2 ==> Σ(n = 1 to ∞) (-1)ⁿ/n², absolutely convergent (via p=2 series).

Hence, the series converges (pointwise) on [-2, 2], and has radius of convergence 2.

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Next, we check for uniform convergence on [-2, 2].

Note that for all x in [-2, 2], |xⁿ/(n²2ⁿ)| ≤ 2ⁿ/(n²2ⁿ) = 1/n².

Since Σ(n = 1 to ∞) 1/n² is a convergent p-series, we conclude that

Σ(n = 1 to ∞) xⁿ/(n²2ⁿ) is uniformly convergent by the Weierstrauss M-Test.

Moreover, since g_n (x) = xⁿ/(n²2ⁿ) is continuous on [-2, 2] for all n, we conclude (with the previous remark) that Σ(n = 1 to ∞) xⁿ/(n²2ⁿ) is continuous on [-2, 2].

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I hope this helps!