Exam 11: Taylor Polynomials and Infinite Series

The Bessel function f(x) of order zero has the Taylor series at x = 0 given by $f ( x ) = 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 64 } - \frac { x ^ { 6 } } { 2304 } + \ldots$ What is $\mathrm { f } ( 4 )$ (0) ?

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The fifth Taylor polynomial of f(x) at x = 0 is $p 5 ( x ) = x ^ { 4 } - 7 x ^ { 3 } + 2$ .

A polynomial f(x) of degree 3 for which f(1) = -1, f'(1) = 2, f''(1) = -1, and f'''(1) = -2 is given by

Determine the sum of the following geometric series: $\frac { 27 } { 5 }$ + $\frac { 18 } { 5 }$ + $\frac { 12 } { 5 }$ + $\frac { 8 } { 5 }$ + ... . Enter a reduced fraction of form $\frac { a } { b }$ .

The Newton-Raphson algorithm is applied to estimate $\sqrt { 10 }$ . If $x 0$ = 3, find $x _ { 2 }$ .

Find the Taylor Series at x = 0 for f(x) = $e ^ { x + 1 }$ .

The Taylor Series for $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ at x = 0 is given by $f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 } + \ldots$ . Find $f ( 3 ) ( 0 )$

Determine the third Taylor polynomial of f(x) = $x ^ { 3 }$ - 3x at x = 0. Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Use two repetitions of the Newton-Raphson algorithm to approximate the zero of f(x) = sin x - cos x near x = 0. Enter just a real number rounded off to two decimal places.

Determine the sum of the following geometric series: 1 - $\frac { 1 } { 2 ^ { 3 } }$ + $\frac { 1 } { 2 ^ { 6 } }$ - $\frac { 1 } { 2 ^ { 9 } }$ + $\frac { 1 } { 2 ^ { 12 } }$ - ... . Enter just a reduced fraction of form $\frac { a } { b }$ .

Find the Taylor series at x = 0 of f(x) = $x ^ { 2 }$ $\mathrm { e } ^ { 2 x }$ . Use enough terms to calculate 0.25e to two decimal places of accuracy. Enter just a real number to 2 decimal places.

Solve the problem. -England decides to decrease its taxes by £7 billion. It is estimated that of each pound received, a typical citizen will spend 90%. The level of economic activity generated by the tax cut is therefore estimated to be: $7 \cdot ( 0.90 ) + 7 \cdot ( 0.90 ) ^ { 2 } + 7 \cdot ( 0.90 ) ^ { 3 } + \ldots$ billion pounds. This amount is equal to what?

Let f(x) = $e ^ { x / 2 }$ . Determine the second Taylor polynomial of f(x) at x = 2.

Which of the following series converge?

Determine the sum of the series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 2 } { e } \right) ^ { n }$ .

Use the Newton-Raphson algorithm with three repetitions to approximate the zero of $f ( x ) = \cos x + x - 2$ near $x = 3$ Enter just a real number rounded off to two decimal places.

Determine the sum of the infinite series: $\sum _ { k = 0 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { 2 k }$ .

Write down the fourth Taylor polynomial of f(x) = $e ^ { - x ^ { 2 } }$ at x = 0. Enter your answer an an unlabeled polynomial in x in standard Taylor polynomial form (i.e., constant first, highest power last ).

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0.19 \overline { 696 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Is this the graph of y = $e ^ { x }$ and are its first two Taylor polynomials at x = 0 on the same axis?