Test # 3

Find the area under the curve \(f(x)=x^3\) on \(x=[-1.5,4.5]\) using the Midpoint Rule with \(n=6\)

Use Simpson's rule with n = 4 to approximate \(\displaystyle\int_1^3\frac{\sin(x)}{x}\,dx\).

\(\displaystyle\int_1^3\frac{\sin(x)}{x}\,dx\approx\)

Keep at least 2 decimal places accuracy in your final answer

t week	0	5.5	11	13.5	19	20
r(t) words per week	12	12	12.25	12.5	14	12

Ian is trying to learn new Spanish words every week. The function \(r(t)\) in words per week is the rate at which he learns new Spanish words.

A. Use the Trapezoidal Rule with sub-intervals indicated by the table to estimate \(\displaystyle\int_0^{20}r(t)\,dt\).

B. What are the final units on your estimate?

Compute the given integral. Enter the exact value of the answer.

\(\displaystyle\int_{-\infty}^0\frac{1}{x^2+1}\,dx=\)

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as "oo" (without the quotation marks). If it diverges to negative infinity, state your answer as "-oo". If it diverges without being infinity or negative infinity, state your answer as "DNE".

\(\displaystyle\int_3^9\frac{9}{\sqrt[3]{x-3}}\,dx\)

Find a formula for the general term \(a_n\) of the sequence assuming the pattern of the first few terms continues.

\(\displaystyle\left\{\frac{7}{3},\ \frac{7}{9},\ \frac{7}{27},\ \frac{7}{81},\ \frac{7}{243},\ \ldots\right\}\)

Assume the first term is \(a_1\)

\(a_n=\)

Find the limit of the sequence:

\(\displaystyle a_n=\frac{4n^2+7n+1}{2n^2+8n+9}\)

Evaluate the telescoping series or enter DNE if it diverges.

\(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^5}-\frac{1}{(n+1)^5}\ =\)

Determine the sum of the following series.

\(\displaystyle\sum_{n=1}^{\infty}\left(\frac{3^n+5^n}{9^n}\right)\)

Consider the series \(\displaystyle\sum_{n=1}^{\infty}\ln\left(\frac{3n+3}{4n+2}\right)\)

Based on the Divergence Test, does this series Diverge?

Inconclusive Diverges

Find the value of

\(\displaystyle\int_2^{\infty}\frac{dx}{(7x-2)^6}\)

Determine whether

\(\displaystyle\sum_{n=2}^{\infty}\frac{1}{(7n-2)^6}\)

is a convergent series. Enter C if series is convergent, D if series is divergent.

To test the series \(\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}\) for convergence, you can use the P-test. (You could also use the Integral Test, as is the case with all series of this type.) According to the P-test:

\(\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}\) diverges the P-test does not apply to \(\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}\) \(\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}\) converges

Now compute \(s_4\), the partial sum consisting of the first 4 terms of \(\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}\):

\(s_4=\)

To test \(\displaystyle\sum_{n=1}^{\infty}\frac{n^7}{n^8-5}\) for convergence you could compare it to \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\) and use the limit comparison test.

Begin that process by computing the limit below.

\(\displaystyle\lim_{n\to\infty}\frac{\dfrac{n^7}{n^8-5}}{\dfrac{1}{n}}=\)

Based on this limit and what you know about \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\), what does the limit comparison test tell you about \(\displaystyle\sum_{n=1}^{\infty}\frac{n^7}{n^8-5}\) ?

The series Converges The series Diverges The test is Inconclusive

Which is the best description for the series?

\(\displaystyle\sum_{n=1}^{\infty}\frac{8n^4}{n^6-9}\)

Diverges by the Divergence Test (nth term test). Convergent Geometric series. Divergent Geometric series. Divergent Harmonic series. Convergent Alternating Harmonic Series. Convergent p-series. Divergent p-series. Convergent by Comparison/Limit Comparison Test. Divergent by Comparison/Limit Comparison Test. Convergent by Alt. Series Test. Convergent by Ratio/Root Test. Divergent by Ratio/Root Test.

Does the series \(\displaystyle\sum_{k=2}^{\infty}(-1)^k\,\frac{\ln(k^5)}{k^7+9}\) converge absolutely, converge conditionally or diverge?

diverges converges conditionally converges absolutely

Consider the general alternating harmonic series: \(\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{7n+2}\)

Determine whether the series converges absolutely, conditionally, or diverges.

converges absolutely diverges converges conditionally

Let us use the ratio test to determine whether or not the following series converges:

\(\displaystyle\frac{1}{8}+\frac{2}{15}+\frac{3}{11}+\cdots+\frac{n!}{7n+1}+\cdots\)

(a) Let \(a_n\) be the \(n^{\text{th}}\) term of the series. Then the ratio \(\displaystyle\left|\frac{a_{n+1}}{a_n}\right|\) in terms of \(n\), after simplifying, is

(b) So \(\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\)

(Enter oo for infinity, -oo for negative infinity, DNF if it does not exist, or the numerical value if it exists.)

(c) Therefore,

the series converges. the series diverges.

Test the series below for convergence using the Ratio Test.

\(\displaystyle\sum_{n=1}^{\infty}\frac{n+5}{3^{4n+1}}\)

The limit of the ratio test simplifies to \(\displaystyle\lim_{n\to\infty}|f(n)|\) where

\(f(n)=\)

The limit is:

(enter oo for infinity if needed)

Based on this, the series

Test the series below for convergence using the Root Test.

\(\displaystyle\sum_{n=1}^{\infty}\left(\frac{5n}{6n+2}\right)^n\)

The limit of the root test simplifies to \(\displaystyle\lim_{n\to\infty}|f(n)|\) where

\(f(n)=\)

The limit is:

(enter oo for infinity if needed)

Based on this, the series

Converges Diverges

Which of the following sequences are convergent sequences? Select all that apply.

\(\displaystyle a_n=\frac{1}{4}\) \(\displaystyle a_n=\frac{1}{4}+(-1)^n\) \(a_n=4\) \(\displaystyle a_n=\frac{1}{n}\) \(\displaystyle a_n=\frac{\cos(n)}{n}\) \(\displaystyle a_n=\frac{4^n}{n^4}\)