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# 高等微積分*+_

Prove that each of the following series is uniformly convergent over the set of values of x given:(c) Σ(n=1→ ∞) sin nx/n^2+1,all x (d) Σ(n=1→ ∞) e^nx/2^n, x<=log3/2.

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- Scharze spaceLv 71 decade agoFavorite Answer
(c) |sin(nx)/(n^2+1)|<=1/n^2 ,Σ1/n^2 convergent

By M-test

Σsin(nx)/(n^2+1) is uniformly convergent

(d) 0<e^(nx)/2^n<=e^(ln(3/2)^n)/2^n=(3/4)^n

Σ(3/4)^n<+∞

so by M-test

Σe^(nx)/2^n uniformly convergent

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