Question

Entered [(-1)^n]([x^(2^n+5)]/(2n+1)) [(-1)^n]([x^(2n+6)]/[(2n+6)(2n+1)]) 1.9E-06 2 At least one of the answers above is NOT correct. The power series centered at 0 for the inverse tangent is tan^-1x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}. Use this series to approximate the definite integral with |error| ? 0.000004. \int_0^{1/3} x \tan^{-1}(x) \, dx = \int_0^{1/3} \sum_{n=0}^{\infty} ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) \, dx = \sum_{n=0}^{\infty} ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) \Big|_0^{1/3} \approx 0.0000019 How many terms are needed to reach this approximation? 2 Answer Preview Result (-1)^n \frac{x^{2n+5}}{2n+1} correct (-1)^n \frac{x^{2n+6}}{(2n+6)(2n+1)} correct 1.9 \times 10^{-6} incorrect 2 correct

          Entered
[(-1)^n]*([x^(2^n+5)]/(2*n+1))
[(-1)^n]*([x^(2n+6)]/[(2*n+6)*(2*n+1)])
1.9E-06
2
At least one of the answers above is NOT correct.
The power series centered at 0 for the inverse tangent is
tan^-1x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}.
Use this series to approximate the definite integral with |error| ? 0.000004.
\int_0^{1/3} x \tan^{-1}(x) \, dx = \int_0^{1/3} \sum_{n=0}^{\infty} ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) \, dx = \sum_{n=0}^{\infty} ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) \Big|_0^{1/3} \approx 0.0000019
How many terms are needed to reach this approximation? 2
Answer Preview
Result
(-1)^n \frac{x^{2n+5}}{2n+1}
correct
(-1)^n \frac{x^{2n+6}}{(2n+6)(2n+1)}
correct
1.9 \times 10^{-6}
incorrect
2
correct

Entered
[(-1)^n]*([x^(2^n+5)]/(2*n+1))
[(-1)^n]*([x^(2n+6)]/[(2*n+6)*(2*n+1)])
1.9E-06
2
At least one of the answers above is NOT correct.
The power series centered at 0 for the inverse tangent is
tan^-1x = ∑n=0^∞ ((-1)^n x^2n+1)/(2n+1).
Use this series to approximate the definite integral with |error| ? 0.000004.
∫0^1/3 x tan^-1(x) dx = ∫0^1/3 ∑n=0^∞ ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) dx = ∑n=0^∞ ((-1)^n(x^(2n+6))/((2n+6)(2n+1))) |0^1/3 ≈0.0000019
How many terms are needed to reach this approximation? 2
Answer Preview
Result
(-1)^n (x^2n+5)/(2n+1)
correct
(-1)^n (x^2n+6)/((2n+6)(2n+1))
correct
1.9 ×10^-6
incorrect
2
correct

Added by Rocio S.

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