The use of math tags around this note triggers MathJax to…
The use of math tags around this note triggers MathJax to display the LaTex written on this page as MathML objects. You must show all of the work you did that leads you to your answer. Be sure to follow the instructions on each problem. 1.) Find the following integral using the methods from our course. \[\int \frac{4x-1}{\left(x-3\right)\left(x+1\right)}\,dx\] 2.) Find the following integral using the methods from our course. \[\int \frac{5}{\left(x-2\right)\left(x^{2}+4\right)}\,dx\] 3.) Use \(S_{6}\), Simpson’s Rule with \(6\) subintervals, to approximate the following integral: \[\int_{2}^{5}\frac{3}{x^{3/2}+1}\,dx\] 4.) Find the value of the following improper integral by first converting it to the appropriate limit. \[\int_{3}^{\infty}\frac{2}{\left(x-1\right)^{3}}\,dx\] 5.) Find the value of the following improper integral by first converting it to the appropriate limit. \[\int_{1}^{3} \frac{1}{\sqrt[4]{x-1}}\,dx\] 6.) Determine whether the following sequence converges or diverges. If it converges, find the limit. Show all of your work and/or reasoning. Note: This is just a sequence, not a series. \[a_{n}=\frac{2n^{3}-5n}{6-5n^{3}}\] 7.) Find the value of the following series. \[\sum_{n=0}^{\infty}\frac{3\times 5^{n}}{2^{3n+2}}\] 8.) Find the value of the following series. \[\sum_{n=1}^{\infty}\frac{3}{n\left(n+2\right)}\] 9.) Let \(S=\sum_{n=1}^{\infty}\frac{1}{n^{2}}\). First calculate \(S_{5}\), the fifth partial sum. Then use the error estimate for the integral test to find a two-sided estimate for the error, \(R_{5}\). That is, you will have a result like: \[a < R_{5} < b\] 10.) For the following series, determine whether they converge or diverge, stating your reasoning and which test you are using. (This will not be a long explanation.) All summations are \(n=1\) to \(\infty\). Comparison and ratio/root tests may not be used. a.) \(\sum 3n^{-3/2}\) b.) \(\sum \frac{5}{n}\) c.) \(\sum \frac{n+1}{n^{3}}\) Hint: Break into two pieces first, then analyze each piece. Extra Credit. [5 points] 11.) Find the limit of the following sequence, explaining your reasoning. (An answer alone will receive no credit.) \[\lim_{n\to\infty}\frac{10+3\cos\left(n^{2}+1\right)}{n}\]
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