Problem 1
Use the Limit Comparison Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{n}{n^{2}+2 n+3} $$
Problem 1
Find the Maclaurin polynomial of order 4 for \(f(x)\) and use it to approximate \(f(0.12) .\) $$ f(x)=e^{2 x} $$
Problem 1
An explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n}{3 n-1} $$
Problem 1
In Problems \(1-14\), indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series $$ \sum_{k=1}^{\infty}\left(\frac{1}{7}\right)^{k} $$
Problem 1
Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=0}^{\infty} \frac{1}{k+3} $$
Problem 1
Find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{(n-1) !} $$
Problem 1
Show that each alternating series converges, and then estimate the error made by using the partial sum \(S_{9}\) as an approximation to the sum \(S\) of the series (see Examples \(1-3)\). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2}{3 n+1} $$
Problem 1
find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series. $$ f(x)=\frac{1}{1+x} $$
Problem 2
Use the Limit Comparison Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{3 n+1}{n^{3}-4} $$
Problem 2
An explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{3 n+2}{n+1} $$
