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AP Calculus BC : Taylor Series

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Example Questions

Example Question #61 : Polynomial Approximations And Series

Find the interval of convergence of for the series $\sum_{n=1}^{\infty }\frac{3^nx^n}{n!}$ .

Possible Answers:

$\left [ \frac{-1}{3}, \frac{1}{3} \right ]$

$\left ( -\infty , \infty \right )$

$\left ( \frac{-1}{3}, \frac{1}{3} \right )$

$\left ( -3, 3\right )$

(-1,0)

Correct answer:

$\left ( -\infty , \infty \right )$

Explanation:

Using the root test,

$\lim_{n\rightarrow \infty }\left | \frac{a_{n+1}}{a_n} \right |=\lim_{n\rightarrow \infty }\left |\frac{3^{n+1} x^{n+1}}{(n+1)!} \cdot \frac{n!}{3^nx^n} \right |=\left | 3x\right |\lim_{n\rightarrow \infty }\left | \frac{1}{n+1} \right |=0$

Because 0 is always less than 1, the root test shows that the series converges for any value of x.

Therefore, the interval of convergence is:

$(-\infty, \infty)$

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Example Question #61 : Polynomial Approximations And Series

Find the interval of convergence for of the Taylor Series $\sum_{n=0}^{\infty }n!(x-5)^n$ .

Possible Answers:

$\left ( \frac{-1}{5}, \frac{1}{5} \right )$

x=5

$\left ( -5, 5\right )$

x=0

$(-\infty, \infty)$

Correct answer:

x=5

Explanation:

Using the root test

$\lim_{n\rightarrow \infty }\left | \frac{a_{n+1}}{a_n} \right |=\lim_{n\rightarrow \infty }\left |\frac{(n+1)!(x-5)^{n+1}}{n!(x-5)^n} \right |=\left |x-5 \right |\lim_{n\rightarrow \infty }\left | n+1 \right |$

and

$\lim_{n\rightarrow \infty }\left | n+1 \right |=\infty$ . T

herefore, the series only converges when it is equal to zero.

This occurs when x=5.

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Example Question #62 : Taylor Series

Let $P_{5}(x)$ be the fifth-degree Taylor polynomial approximation for f(x) = sin(x) , centered at x = 0 .

What is the Lagrange error of the polynomial approximation to sin(1) ?

Possible Answers:

$\frac{1}{6!}$

$\frac{1}{5!}$

$\frac{1}{7!}$

Correct answer:

$\frac{1}{7!}$

Explanation:

The fifth degree Taylor polynomial approximating sin(x) centered at x=0 is:

$sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!}$

The Lagrange error is the absolute value of the next term in the sequence, which is equal to $\left |\frac{x^7}{7!} \right |$ .

We need only evaluate this at and thus we obtain $\frac{1^7}{7!} = \frac{1}{7!}$

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AP Calculus BC : Taylor Series

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #61 : Polynomial Approximations And Series

Example Question #61 : Polynomial Approximations And Series

Example Question #62 : Taylor Series

All AP Calculus BC Resources

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