Exam 11: Taylor Polynomials and Infinite Series

Determine the sum of the following infinite series: $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } + ( - 1 ) ^ { n } } { 3 n }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0 . \overline { 498 }$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

Find an infinite series that converges to the value of $\int _ { 0 } ^ { 1 } 2 x e ^ { - x }$ dx. Is $1 - \frac { 2 } { 3 } + \frac { 1 } { 4 } - \frac { 1 } { 15 } \ldots$ correct?

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

Determine the sum of the following geometric series: 3 - 1.8 + 1.08 + .648 - ... . Enter just a real number rounded off to three decimal places.

Determine the sum of the series $\mathrm { e } ^ { - 1 }$ + $e ^ { - 2 }$ + $\mathrm { e } ^ { - 3 }$ + ... if it converges.

Find the third Taylor polynomial of f(x) = $x ^ { 2 }$ + sin x at x = 0. Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \cos 3 x + \sin 2 x$ . Is $1 + 2 x - \frac { 9 x ^ { 2 } } { 2 ! } - \frac { 8 x ^ { 3 } } { 3 ! } + \ldots$ the correct answer?

Suppose f(x) = $x ^ { 4 }$ - 7 $x ^ { 3 }$ + 2. The third Taylor polynomial of f(x) at x = 0 is $\mathrm { P } 3 ( x ) = 2 - 7 x ^ { 3 }$ .

Determine the first four non-zero terms of the Taylor series at x = 0 for f(x) = sin $x ^ { 3 }$ . Is $f ( x ) = x ^ { 3 } + \frac { x ^ { 9 } } { 3 ! } + \frac { x ^ { 15 } } { 5 ! } + \frac { x ^ { 21 } } { 7 ! }$ the correct answer?

Find the third Taylor polynomial of f(x) = sin x at x = 0 and use it to approximate $\sin \frac { 1 } { 2 }$ . Enter just a real number rounded off to two decimal places.

The area of a circle with radius 1 is π. If f(x) = $\sqrt { 1 - x ^ { 2 } }$ gives the top half of this circle, as illustrated below, use the second Taylor polynomial of f(x) at x = 0 to find an approximate value for π. Is the following correct? 2()=1- \approx 1- = so \pi\approx

Determine the sum of the following geometric series: $1 + ( 0.25 ) ^ { 2 } + ( 0.25 ) ^ { 4 } + ( 0.25 ) ^ { 8 } + \ldots$ . Enter a reduced fraction of form $\frac { a } { b }$ .

Find the Taylor series expansion for f(x) = $\frac { x } { 1 - x }$ and use it to determine which of the following is false?

Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \ln ( x + 1 )$ . Is $f ( x ) = x + \frac { x ^ { 2 } } { 2 ! } + \frac { 2 x ^ { 3 } } { 3 ! } + \frac { 6 x ^ { 4 } } { 4 ! }$ correct?

Use the integral test to determine whether the infinite series $\sum _ { k = 1 } ^ { \infty } \frac { 5 k ^ { 2 } + 1 } { 2 k ^ { 3 } - 1 }$ is convergent or divergent. Enter just "convergent" or "divergent".

Estimate $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ by using the second Taylor polynomial for f(x) = $e ^ { x ^ { 2 } }$ . Is $\int _ { 0 } ^ { 1 } \mathrm { e } ^ { \mathrm { x } ^ { 2 } } \mathrm { dx } \approx \frac { 4 } { 3 }$ the solution?

Determine the sum of the geometric series $\frac { 2 ^ { 2 } } { 5 ^ { 2 } }$ - $\frac { 2 ^ { 3 } } { 5 ^ { 3 } }$ + $\frac { 2 ^ { 4 } } { 5 ^ { 4 } }$ - ..., if it is convergent.

Determine the sum of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 - 2 ^ { n } } { 3 n }$ .

Find the third Taylor polynomial of f(x) = cos x at x = $\frac { \pi } { 2 }$ . Enter your answer as an unlabeled polynomial in x - $\frac { \pi } { 2 }$ in standard form (i.e., highest powers first).