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Problem 43 Hard Difficulty

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

$ f(x) = \sin^2 x $ [Hint: Use $ \sin^2 x = \frac {1}{2} ( 1 - \cos 2x). $ ]

Answer

$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{2 n-1} x^{2 n}}{(2 n) !}, \quad R=\infty$

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Video Transcript

All right, so we can expend this function at the exit, sign a square and the using this hint. All right, so after equals, two, one half over one half times, cousin of two ants. Okay, we're gonna expand, Chosen to X, which is very familiar of us. So one half minus one half times the experiment is going to be one minus two X cube or Victoria plus to excess to hell before or for Victorio. And, of course, it's going to be two extra power for six over six Victoria. And so long. And what is it? You two, which is equal to one half minus one half times so signal. And it's become two to the power to extradite orthe two in over two in factorial. Of course, it's from zero to many. And hear him honest One to power and minus one. Look, actually, basalt. So when in this zero and it becomes too when an issue Oh, no. This is actually could horror in the not minus one minus one. Yeah, this should be correct. We'LL plug it in and equals one. So this is two kings for two is four and three six and Yeah, this is to end and three over. Here it is. Absolute crept end. Obtain the MEChA Lorna series for the function. Okay, so it becomes to ninety one times that you want to our answer me and that you wanted off him. Plus one times So two three power for two months, One extra power to end over to Victorio and the yet for us one half. And we can continue to simplify this. It looks too. And from zero, it's one so infinitely so let you want off this one Two powerful twin minus one times x to in over tutorial plus one half Plus when the zero And when in this zero it becomes too knotty One times that you want times we'LL have. So it's not one half. Why this one half? Yeah, And this kind of hot. So our final answer What is this? Siri's

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