Anonymous
Anonymous asked in 科學數學 · 1 decade ago

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.

1 Answer

Rating
  • 1 decade ago
    Favorite 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

Still have questions? Get your answers by asking now.