integral (1-cos x)/x dx . limits 0 to 1/3. use power series to estimate the integral to within .001
i tried to solve it and i am kinda stuck on the last part. i got
x^2/4 - x^4/(4)(4!) + x^6/ (6)(6!) from 0 to 1/3 to estimate the error do i just plug in 1/3 for x?
i tried to solve it and i am kinda stuck on the last part. i got
x^2/4 - x^4/(4)(4!) + x^6/ (6)(6!) from 0 to 1/3 to estimate the error do i just plug in 1/3 for x?
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Yep. You get a series for your integrand, then integrate and evaluate. Your series is alternating, so the magnitude of the error is bounded above by the absolute value of the first term "left out".
Here the next term would be x^8 / (8 * 8!) which at 1/3 is already much smaller than your required error bound.
Actually the third term is already 0.0001286008230452674897119341563786 so the first two terms are enough to get you your desired error bound.
The exact value of the integral is close to 0.0276494.. and the first two terms return 0.027649177...
So on a test or something where you're trying to demonstrate understanding of the material, I would use only the first two terms.
Here the next term would be x^8 / (8 * 8!) which at 1/3 is already much smaller than your required error bound.
Actually the third term is already 0.0001286008230452674897119341563786 so the first two terms are enough to get you your desired error bound.
The exact value of the integral is close to 0.0276494.. and the first two terms return 0.027649177...
So on a test or something where you're trying to demonstrate understanding of the material, I would use only the first two terms.