Exam 11: Taylor Polynomials and Infinite Series

The Taylor series at x = 0 for f(x) = ln ( $\frac { 1 + x } { 1 - x }$ ) is 2x + $\frac { 2 } { 3 }$ $x ^ { 3 }$ + $\frac { 2 } { 5 }$ $x ^ { 5 }$ + $\frac { 2 } { 7 }$ $x ^ { 7 }$ + ..., $| x |$ < 1. Find $f ( 6 )$ (0). Enter just an integer.

The Newton-Raphson algorithm is used to approximate the zero of f(x) = $x ^ { 3 }$ + x - 5 between x = 1 and x=2 . If x₀ = 1, find x₁.

Determine the sum of the following infinite series: $\sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k }$ $\frac { 2 } { 7 \mathrm { k } }$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

The Newton-Raphson algorithm is applied to estimate a zero of f(x) with $x 0$ = 3. Which of the following statements is true?

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = e ^ { x }$ . Is $f ( x ) = 1 - x + \frac { x ^ { 2 } } { 2 ! } - \frac { x ^ { 3 } } { 3 ! }$ correct?

Find the first three non-zero terms of the Taylor series at x = 0 of $f ( x ) = e ^ { x } \sin x$ . Is $f ( x ) = x + x ^ { 2 } + \frac { x ^ { 3 } } { 3 }$ correct?

Find the second Taylor polynomial of f(x) = $\sqrt { x }$ at x = 9 and use it to approximate $\sqrt { 9.1 }$ . Enter just a real number rounded off to two decimal places.

Let $x _0$ = 2. Use three repetitions of the Newton-Raphson algorithm to approximate $\sqrt [ 3 ] { 5 }$ . Enter just a real number rounded off to two decimal places (no label).

Determine the sum of the following geometric series: 1 + $\frac { 1 } { \sqrt { 2 } }$ + $\frac { 1 } { 2 }$ + ... . Enter your answer exactly in the form $\frac { \sqrt { a } } { \sqrt { a } - b }$ .

Use the comparison test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ^ { 3 } + k - 1 }$ is convergent or divergent. Enter just "convergent" or "divergent".

Find the Taylor Series at x = 0 for f(x) = $\frac { \sin x } { x }$ .

Below is a graph of function f(x). Which of the following could be the second Taylor polynomial of f(x) at $x = a ?$

Use the integral test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { ( k + 2 ) ^ { 2 } }$ is convergent or divergent? Enter just the word "convergent" or "divergent".

Below is a graph of the function f(x). If $x 0$ is taken as the initial approximation of the zero of f(x), then which of the following points, A, B, C, or D could be given by the Newton-Raphson algorithm as the next approximation?

Determine the third Taylor polynomial of f(x) = $\frac { 1 } { \sqrt { 1 - x } }$ at x = 0. Enter your answer as an unlabeled polynomial in x in standard form (i.e., highest powers first).

Determine the first three non-zero terms of the Taylor series at x = 0 for $f ( x ) = x \cos x - \sin x$ . Is $\frac { ( 3 ! - 2 ! ) x ^ { 3 } } { 2 ! 3 ! } + \frac { ( 5 ! - 4 ! ) x ^ { 5 } } { 4 ! 5 ! } + \frac { ( 7 ! - 6 ! ) x ^ { 7 } } { 6 ! 7 ! }$ the correct answer?

Determine the sum of the series $\sum _ { n = 2 } ^ { \infty } \left( - \frac { 1 } { 2 } \right) ^ { n }$

Determine the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { 4 ^ { n - 1 } }$ .

Determine the third Taylor polynomial of f(x) = ln(2 - x) at x = 1 and use it to estimate $\ln ( 1.3 ) .$ [Hint: $\ln ( 1.3 ) = f ( 0.7 )$ .] Enter your answer as an unlabeled polynomial in $x - 1$ in standard form : $\ln ( 1.3 ) \approx p _ { 3 } ( 0.7 ) = 0.264$

Determine the sum of the infinite geometric series $\sum _ { \mathrm { k } = 0 } ^ { \infty } 3 ^ { - \mathrm { k } } \ln 2$ Enter your answer exactly in the reduced form $\frac { a \ln b } { c }$ .