Write the Maclaurin series for e^x up to and including the x^4 term.

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Multiple Choice

Write the Maclaurin series for e^x up to and including the x^4 term.

The idea being tested is the Maclaurin (Taylor) series for e^x, which uses that all derivatives of e^x are e^x and at 0 each derivative evaluates to 1, so the coefficient of x^n is 1/n!. Building up to the x^4 term gives 1 + x + x^2/2! + x^3/3! + x^4/4! = 1 + x + x^2/2 + x^3/6 + x^4/24. The other forms misplace or misstate these coefficients (for example, one would require an x^5 term, another has x^2 with coefficient 1, and another uses x^3/3 instead of 1/6), so they don’t match the correct factorial pattern.

The idea being tested is the Maclaurin (Taylor) series for e^x, which uses that all derivatives of e^x are e^x and at 0 each derivative evaluates to 1, so the coefficient of x^n is 1/n!. Building up to the x^4 term gives 1 + x + x^2/2! + x^3/3! + x^4/4! = 1 + x + x^2/2 + x^3/6 + x^4/24. The other forms misplace or misstate these coefficients (for example, one would require an x^5 term, another has x^2 with coefficient 1, and another uses x^3/3 instead of 1/6), so they don’t match the correct factorial pattern.

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