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Math Help - Show that sin(a_n) converges

  1. #1
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    Show that sin(a_n) converges

    Show that if \sum{a_n} is a convergent positive-term series, then the series \sum{sin(a_n)} also converges.
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    Just notice that if 0\le x then \sin(x)\le x.
    Use the basic comparison test.
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    Quote Originally Posted by Plato View Post
    Just notice that if 0\le x then \sin(x)\le x.
    Use the basic comparison test.
    but \sin a_n is not necessarily positive! we can remove this difficulty by using the inequality |\sin x| \leq |x|.

    so if \sum a_n is convergent with a_n \geq 0, then \sum \sin a_n is absolutely convergent and hence convergent.
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    Quote Originally Posted by NonCommAlg View Post
    but \sin a_n is not necessarily positive! we can remove this difficulty by using the inequality |\sin x| \leq |x|.
    Actually \sin(a_n)>0 for almost all n.
    It was given that  (a_n) is a positive sequence and we know that  (a_n)\to 0 .
    Therefore, for almost all n we have 0< a_n<\frac{\pi}{2} which implies 0<\sin(a_n)<a_n.
    So you are mistaken. Comparison works.

    P.S. If 0<x then \sin(x)<x.
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    Quote Originally Posted by Plato View Post
    Actually \sin(a_n)>0 for almost all n.
    It was given that  (a_n) is a positive sequence and we know that  (a_n)\to 0 .
    Therefore, for almost all n we have 0< a_n<\frac{\pi}{2} which implies 0<\sin(a_n)<a_n.
    So you are mistaken. Comparison works.

    P.S. If 0<x then \sin(x)<x.
    you're right. i ignored the fact that a_n \to 0.
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