Using the Maclaurin series for e^x truncated after the x^3 term, approximate e^1.

Using the Maclaurin series for e^x truncated after the x^3 term, approximate e^1.

• A. 8/3
• B. 2
• C. e
• D. 9/4

Answer: (A)

Using the Maclaurin series for e^x, we expand e^x as 1 + x + x^2/2! + x^3/3! + ... . Truncating after the x^3 term means we keep only 1 + x + x^2/2 + x^3/6. Plugging in x = 1 gives 1 + 1 + 1/2 + 1/6 = 8/3, which is about 2.6667. The true value of e^1 is a bit larger because the next term 1/24 would be added if we continued, yielding e ≈ 2.71828. So the best approximation from this truncation is 8/3.

Using the Maclaurin series for e^x, we expand e^x as 1 + x + x^2/2! + x^3/3! + ... . Truncating after the x^3 term means we keep only 1 + x + x^2/2 + x^3/6. Plugging in x = 1 gives 1 + 1 + 1/2 + 1/6 = 8/3, which is about 2.6667. The true value of e^1 is a bit larger because the next term 1/24 would be added if we continued, yielding e ≈ 2.71828. So the best approximation from this truncation is 8/3.