Solve the problem.
-You plan to estimate e by evaluating the Maclaurin series for $f ( x ) = e ^ { x }$ at $x = 1$. How many terms of the series would you have to add to be sure the estimate is good to 5 decimal places?
A) 13
B) 10
C) 11
D) 12

Question

Quizplus · Accepted Answer

To estimate $e$ to 5 decimal places accurately using the Maclaurin series for $f(x) = e^x$ at $x = 1$, we need to consider the error term in the Taylor series approximation. The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. The error term for the Taylor (or Maclaurin) series is given by $\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$ for some $c$ in the interval [0, x], where $f^{(n+1)}$ is the $(n+1)$th derivative of $f$. Since all derivatives of $e^x$ are $e^x$ and $e^1 = e$, the error term for estimating $e$ at $x=1$ is less than or equal to $\frac{e}{(n+1)!}$. To ensure the error is less than $0.5 \times 10^{-5}$ (for 5 decimal places accuracy), we find the smallest $n$ such that $\frac{e}{(n+1)!} < 0.5 \times 10^{-5}$. Since $e < 3$, it suffices to find $n$ for which $\frac{3}{(n+1)!} < 0.5 \times 10^{-5}$. Testing values, we find that $n=10$ (thus using 11 terms) satisfies this condition, ensuring an estimate good to 5 decimal places.

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