Exam 11: Taylor Polynomials and Infinite Series

The geometric series 1 + $( 0.2 ) ^ { 3 }$ + $( 0.2 ) ^ { 6 }$ + $( 0.2 ) ^ { 9 }$ + ... (I) converges (II) is equal to $\sum _ { k = 0 } ^ { \infty } \left( \frac { 1 } { 125 } \right) ^ { k }$ (III) is equal to $\sum _ { k = 0 } ^ { \infty } ( 0.2 ) ^ { 3 k }$

Suppose the second Taylor polynomial for f(x) at x = 3 is $P 2 ( x ) = 2 ( x - 3 ) - \frac { 1 } { 3 } ( x - 3 ) ^ { 2 }$ . Find f''(3). Enter just a reduced fraction.

A student receives $1000 at the start of each month from his parents. Every month the student spends 70% of all the money he has. If the only money the student receives is the money from his parents, estimate how much money the student will have at the beginning of each month after an extended period of time. Enter just a reduced fraction of form $\frac { a } { b }$ .

Find the Taylor series at x = 0 of the function f(x) = $\frac { 1 } { 1 - 3 x }$ by computing three or four derivatives and using the definition of the Taylor series.

Find the Taylor series expansion at x = 0 of $\int 2 x e ^ { - x }$ dx. Is $\left[ x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 } + \frac { 1 } { 4 } x ^ { 4 } + \frac { 1 } { 15 } x ^ { 5 } + \ldots \right] + C$ correct?

If f(x) = 1 - 3(x - 2) + 4 $( x - 2 ) ^ { 2 }$ + 6 $( x - 2 ) ^ { 3 }$ , then what is f''(2)? Enter just an integer.

Use two repetitions of the Newton-Raphson algorithm to find the value of x near zero for which $e ^ { x } = 2 \cos x$ . Enter just a real number rounded off to two decimal places.

Use the Newton-Raphson algorithm with three repetitions to approximate the zero of f(x) = $e ^ { x }$ - 2 near $x = 1$ Enter just a real number rounded off to two decimal places.

Let f(x) = $\frac { 1 } { 1 - x }$ . Determine the fourth Taylor polynomial at x = 0.

Use the integral test to determine whether the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { 2 } { 2 k + 1 }$ is convergent or divergent. Then use the comparison test to determine whether the infinite series $\sum _ { \mathrm { k } = 1 } ^ { \infty } \frac { 4 } { \mathrm { k } + 1 }$ is convergent or divergent. Enter just two words which answer the two questions above in order (separated by a comma) where each word is either "convergent" or "divergent".

Determine the sum of the following series: $10 + 4 + \frac { 8 } { 5 } + \frac { 16 } { 25 } + \ldots$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

Use the Newton-Raphson algorithm with two repetitions to estimate the positive solution of sin x = $\frac { 1 } { 2 }$ x. Use $x_ 0$ = 2. Enter just a real number rounded off to two decimal places.

Let f(x) = $x ^ { 3 }$ - 4x - 1. Which of the following statements is true? (All Taylor polynomials are at x = 0.)

Suppose that the first Taylor polynomial of a function f(x) at x = 0 is $\mathrm { P } 1$ (x) = 2 - 3x. Which of the following could be a graph of f(x) ?

Determine the sum of the series $\frac { 1 } { 2 } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 2 ^ { 3 } } + \ldots$

Determine the first four non-zero terms of the Taylor series at x = 0 for $f ( x ) = x e ^ { ( 1 / 2 ) x }$ . Is x - $\frac { x ^ { 2 } } { 2 }$ + $\frac { x ^ { 3 } } { 2 ^ { 2 } \cdot 2 ! }$ - $\frac { x ^ { 4 } } { 2 ^ { 3 } \cdot 3 ! }$ the correct answer?

Determine the sum of the following infinite series: $\sum _ { k = 0 } ^ { \infty } \left( \frac { 1 } { 3 } \right) ^ { k }$ $( 2 ) ^ { \mathrm { k } + 1 }$ . Enter just an integer.

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

It can be shown that $\int _ { 0 } ^ { \infty } x e ^ { - x } d x$ = 1. Use this fact and the integral test to construct an appropriate convergent infinite series. Is $\sum _ { \mathrm { k } = 0 } ^ { \infty } \mathrm { e } ^ { - \mathrm { k } }$ the correct series?

Determine the sum of the following infinite series: $\sum _ { \mathrm { k } = 0 } ^ { \infty } ( 1 - \sqrt { 2 } ) ^ { \mathrm { k } }$ . Enter your answer exactly in the reduced form $\frac { a } { \sqrt { b } }$ .