Write the Maclaurin series for sin x up to and including the x^5 term.

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Multiple Choice

Write the Maclaurin series for sin x up to and including the x^5 term.

The key idea is that the Maclaurin series for sin x has only odd powers and the signs alternate: sin x = x − x^3/3! + x^5/5! − … Up to the x^5 term, you keep the first three nonzero terms, giving sin x ≈ x − x^3/3! + x^5/5!. Evaluating the factorials, that's x − x^3/6 + x^5/120. This matches the required expression because the x^3 term uses 3! in the denominator with a negative sign, and there are no even powers in the series. The other forms would introduce an even power, use the wrong factorial in the denominator, or have the wrong sign for the x^3 term.

The key idea is that the Maclaurin series for sin x has only odd powers and the signs alternate: sin x = x − x^3/3! + x^5/5! − … Up to the x^5 term, you keep the first three nonzero terms, giving sin x ≈ x − x^3/3! + x^5/5!. Evaluating the factorials, that's x − x^3/6 + x^5/120.

This matches the required expression because the x^3 term uses 3! in the denominator with a negative sign, and there are no even powers in the series. The other forms would introduce an even power, use the wrong factorial in the denominator, or have the wrong sign for the x^3 term.

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