Up: Math 222 Syllabus

Problems on Taylor Series and Geometric Series

Which of the following series are power series in x?

(a)

x - x³ + x⁶ - x¹⁰ + x¹⁵ - ^...

(b)

$\displaystyle {\frac{{1}}{{x}}}$ + $\displaystyle {\frac{{1}}{{x^2}}}$ + $\displaystyle {\frac{{1}}{{x^3}}}$ + $\displaystyle {\frac{{1}}{{x^4}}}$ + ^...

(c)

1 + x + (x - 1)² + (x - 2)³ + (x - 3)⁴ + ^...

(d)

x⁷ + x + 2
(H) Find the first four terms of the Taylor series for $\displaystyle {\frac{{1}}{{\sqrt{1+x}}}}$ about 0.
* The series

1 + $\displaystyle {\frac{{2}}{{1!}}}$ + $\displaystyle {\frac{{4}}{{2!}}}$ + $\displaystyle {\frac{{8}}{{3!}}}$ + ^... + $\displaystyle {\frac{{2^n}}{{n!}}}$ + ^...
is a Taylor series (centered at 0) evaluated at a particular value of x. Assuming that the series converges to the value of the function, find the (exact) sum of the series.
(H) Find the sum of $\displaystyle \sum_{{n=0}}^{{\infty}}$ $\displaystyle {\frac{{3^n+5}}{{4^n}}}$ .
A ball is dropped from a height of 10 feet and bounces. Each bounce is 3/4 of the height of the bounce before.

(a)

Find an expression for the height to which the ball rises after it hits the floor for the nth time.

(b)

Find an expression for the total vertical distance the ball has travelled when it hits the floor for the nth time. (Express your answer without using sigma-notation or ``...''.)

(c)

Show that a ball dropped from a height of h feet reaches the ground in ${\frac{{1}}{{4}}}$ $\sqrt{{h}}$ seconds.

(d)

Show that the ball stops bouncing after

$\displaystyle {\frac{{1}}{{4}}}$ $\displaystyle \sqrt{{10}}$ + $\displaystyle {\frac{{1}}{{2}}}$ $\displaystyle \sqrt{{10}}$ $\displaystyle \sqrt{{\frac{3}{4}}}$ $\displaystyle \left(\vphantom{\frac{1}{1-\sqrt{3/4}}}\right.$ $\displaystyle {\frac{{1}}{{1-\sqrt{3/4}}}}$ $\displaystyle \left.\vphantom{\frac{1}{1-\sqrt{3/4}}}\right)$ seconds,
or approximately 11 seconds.