In the Maclaurin series for sin x, what is the coefficient of x^5?

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Multiple Choice

In the Maclaurin series for sin x, what is the coefficient of x^5?

Explanation:
The key idea is that the Maclaurin series for sin x is an alternating sum of odd-power terms: sin x = x − x^3/3! + x^5/5! − x^7/7! + … Each term has the form (+ or −) x^(2n+1) divided by (2n+1)!. For the x^5 term, n = 2, so the coefficient is (+1)/(5!) = 1/120. Therefore, the coefficient of x^5 is 1/120. The signs and the presence of only odd powers explain why the other options don’t fit.

The key idea is that the Maclaurin series for sin x is an alternating sum of odd-power terms: sin x = x − x^3/3! + x^5/5! − x^7/7! + … Each term has the form (+ or −) x^(2n+1) divided by (2n+1)!. For the x^5 term, n = 2, so the coefficient is (+1)/(5!) = 1/120. Therefore, the coefficient of x^5 is 1/120. The signs and the presence of only odd powers explain why the other options don’t fit.

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