Show LHS = RHS for this sum (to n--> infinity)?

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  • JOHN
    Lv 7
    7 years ago
    Favorite Answer

    Let S(x)=tanx/2 + (1/2)tanx/2^2 + (1/2^2)tanx/2^3 +...

    ∫s(x)dx=2lnsec(x/2) + 2lnsec(x/2^2) + 2lnsec(2/2^3)+..

    =2[ln1/{cos(x/2)cos(x/2^2)...to ∞}=2ln(x/sinx)=2lnx - 2ln(sinx)

    [I have proved in Yahoo Answers about a week ago that

    cos(x/2)cos(x/2^2)cos(x/2^3)...cos(x/2^2)..to infinity is sinx/x.

    [See http://answers.yahoo.com/question/index;_ylt=ArPFV...

    =∫(2/x)dx - ∫2cotx)dx. Hence S(x)=2/x-2cotx.

    [I should introduce an arbitrary constant of integration, C, in line 2 above, and then show that C=0. Let x→0.Then the LHS→0 and the RHS→ln1 (since x/sinx→1) =0. Thus C=0].

    The 2nd part of the question immediately follows on setting x=π/2.

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