009C Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.

Click on the boxed problem numbers to go to a solution.

Problem 1

Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.

a)

a_{n}={\frac {\ln(n)}{\ln(n+1)}}

b)

a_{n}={\bigg (}{\frac {n}{n+1}}{\bigg )}^{n}

Problem 2

For each of the following series, find the sum if it converges. If it diverges, explain why.

a)

4-2+1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{8}}+\cdots

b)

\sum _{n=1}^{+\infty }{\frac {1}{(2n-1)(2n+1)}}

Problem 3

Determine if the following series converges or diverges. Please give your reason(s).

a)

\sum _{n=0}^{+\infty }{\frac {n!}{(2n)!}}

b)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^{+\infty} (-1)^n \frac{1}{n+1}}

Problem 4

a) Find the radius of convergence for the power series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{+\infty} (-1)^n \frac{x^n}{n}.}

b) Find the interval of convergence of the above series.

Problem 5

Find the Taylor Polynomials of order 0, 1, 2, 3 generated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\cos(x)} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{\pi}{4}.}

Problem 6

Find the Taylor polynomial of degree 4 of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\cos^2x} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\frac{\pi}{4}} .

Problem 7

A curve is given in polar coordinates by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r=1+\sin \theta }

a) Sketch the curve.

b) Compute

y'={\frac {dy}{dx}}

.

c) Compute Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y''={\frac {d^{2}y}{dx^{2}}}} .