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DoubleCook

  • 3 years ago

What is the Taylor series for f(x)=sin^2(5x)? Hint: using the identity sin^2 x = 1/2 (1-cos(2x))

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  1. timo86m
    • 3 years ago
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    \[( \left( \sin \left( 5\,a \right) \right) ^{2}+10\,\sin \left( 5\,a \right) \cos \left( 5\,a \right) \left( x-a \right) + \left( -25\, \left( \sin \left( 5\,a \right) \right) ^{2}+25\, \left( \cos \left( 5\,a \right) \right) ^{2} \right) \left( x-a \right) ^{2}-{ \frac {500}{3}}\,\sin \left( 5\,a \right) \cos \left( 5\,a \right) \left( x-a \right) ^{3}+ \left( {\frac {625}{3}}\, \left( \sin \left( 5\,a \right) \right) ^{2}-{\frac {625}{3}}\, \left( \cos \left( 5\,a \right) \right) ^{2} \right) \left( x-a \right) ^{4}+{ \frac {2500}{3}}\,\sin \left( 5\,a \right) \cos \left( 5\,a \right) \left( x-a \right) ^{5}+O \left( \left( x-a \right) ^{6} \right) ) \]

  2. timo86m
    • 3 years ago
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    Something like that?

  3. DoubleCook
    • 3 years ago
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    Mmm what is it using the sigma notation? I was thinking something like \[\sum_{0}^{\infty} (1/2)- \frac{ 1/2 (-1)^k (10x)^{2k} }{ (2k)! }\] but my homework says it's wrong? :/

  4. DoubleCook
    • 3 years ago
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    http://www.webassign.net/userimages/knownseries.jpg?db=v4net&id=154152 I used this

  5. timo86m
    • 3 years ago
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    I am sorry i suck at sigma notation

  6. DoubleCook
    • 3 years ago
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    Oh haha don't worry about it, thanks anyways! :]

  7. EulerGroupie
    • 3 years ago
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    I think the 1/2 and the minus are outside the sigma.

  8. EulerGroupie
    • 3 years ago
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    \[\frac{1}{2}-\frac{1}{2}\sum_{0}^{\infty}\frac{(-1)^{k}(10x)^{2k}}{(2k)!}\]

  9. DoubleCook
    • 3 years ago
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    Oh outside? Great, thanks a lot! :]

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