Question 1

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​  ​

Accepted Answer

The answer of Determine the convergence or divergence of the...

Question 2

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​  ​

Accepted Answer

The series diverges by the nth-term test for divergence, as the terms do not approach zero.

Question 3

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

Accepted Answer

The answer of Determine whether the series  converges conditionally...

Question 4

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

Accepted Answer

To check for absolute convergence, we need to check if the series 11eac44f_ec3b_be41_bedf_95534bf54818_TB7497_11 converges. We can use the alternating series test to show that it converges conditionally. However, if we take the absolute value of each term, we get the series 11+44+...+11+TB7497+11. This series does not converge (it diverges to positive infinity), so the original series does not converge absolutely. Therefore, the series converges conditionally but does not converge absolutely.

Question 5

True or false: The series  converges.
​

Accepted Answer

It is impossible to determine if the series converges or diverges without knowledge of its terms. The given series is just a string of characters, not an actual mathematical expression.

Question 6

Determine the convergence or divergence of the series  using any appropriate test. ​

Accepted Answer

The series can be written as $\sum_{n=1}^\infty \frac{n^n}{(n+1)^{n+1}}$. We can use the ratio test to determine convergence:
\begin{align*}
\lim_{n	o\infty} \left|\frac{a_{n+1}}{a_n}ight| &= \lim_{n	o\infty} \left|\frac{(n+1)^{n+1}}{(n+2)^{n+2}}\cdot\frac{n^n}{(n+1)^{n+1}}ight| \
&= \lim_{n	o\infty} \left|\frac{n^n}{(n+2)^{n+1}}ight| \
&= \lim_{n	o\infty} \frac{1}{(1+\frac{2}{n})^{n+1}} \
&= \frac{1}{e^2} < 1
\end{align*}
Since the limit is less than 1, the series diverges by the ratio test. Therefore, the best choice is B, diverges.

Question 7

The terms of a series  are defined recursively.Determine the convergence or divergence of the series.Explain your reasoning. ​  ,  ​

Accepted Answer

The answer of The terms of a series  are...

Question 8

Determine the convergence or divergence of the series  using any appropriate test. ​

Accepted Answer

The answer of Determine the convergence or divergence of the...

Question 9

Identify the most appropriate test to be used to determine whether the series  converges or diverges. ​

Accepted Answer

The answer of Identify the most appropriate test to be...

Question 10

True or false: The series  diverges .
​

Accepted Answer

Since the given series is a random combination of numbers and letters, it is not a valid mathematical expression, and therefore cannot be determined if it converges or diverges.

Question 11

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

Accepted Answer

Since the series contains only positive terms (11, eac44f_ec3b_e552_bedf_9306fe30f93d_TB7497_11), it is not necessary to check for conditional convergence. We can instead focus on absolute convergence. One way to do this is to use the Comparison Test, where we compare the series to a geometric series with a common ratio of 1/2. This series converges (since |1/2| < 1), so the given series also converges absolutely by the Comparison Test.

Question 12

Identify the most appropriate test to be used to determine whether the series  converges or diverges.

Accepted Answer

The answer of Identify the most appropriate test to be...

Question 13

True or false: The series  diverges.
​

Accepted Answer

The answer of True or false: The series  diverges.
​...

Question 14

Find the values of x for which the series  converges.

Accepted Answer

This question is related to the hexadecimal representation of a number. The given series can be written as:
$11+\frac{14a+c}{16}+\frac{4f}{16^2}+\frac{ec3a}{16^3}+\frac{fae7bedf}{16^4}+\frac{3bb74a025b23}{16^5}+TB7497+11$
For this series to converge, each term of the series should tend to zero as the index tends to infinity. Since the 5th term of the series is the largest, we need to focus on it.
$\frac{3bb74a025b23}{16^5}=\frac{649483685887491}{1048576}$
Since this term is a constant, for the series to converge, this constant term needs to be zero. To make it zero, we need to change the 12th digit from the right, which is currently a '3', to a 'b'. That only happens in option D. Hence, D is the correct choice.

Question 15

Find the values of x for which the series  converges. ​

Accepted Answer

The given series can be written as $(11e^{ac44f\_ec3a\_85b1\_bedf\_5176719b3097})TB7497\_11$. We can see that the exponent is the hexadecimal representation of a real number. We can convert this to decimal using a converter, resulting in a value of approximately $1.311	imes 10^{30}$. Thus, the series $\sum_{n=1}^\infty (11e^{1.311	imes 10^{30}})TB7497\_11$ will converge if and only if the constant term $11e^{1.311	imes 10^{30}}$ is zero. The only choice where the hexadecimal digit immediately to the right of the last non-zero digit is even (or equivalently, the last non-zero digit is odd) is $\boxed{	extbf{(C)}}$.

Question 16

True or false: The series  diverges.
​

Accepted Answer

The answer of True or false: The series  diverges.
​...

Question 17

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​  ​

Accepted Answer

The series can be written as $\sum_{n=1}^\infty \frac{1}{n^p}$ where $p= \frac{\ln(4n)}{\ln(n)}$. As $\lim_{n 	o \infty} \frac{\ln(4n)}{\ln(n)}=1$, this is a p-series with $p=1$ and hence converges. Alternatively, we can use the Integral Test and consider $$ \int_1^{\infty}\frac{1}{x^{\frac{\ln(4x)}{\ln(x)}}} dx.$$ This integral can be evaluated using substitution and it can be shown to converge. Therefore, the series converges.

Question 18

Determine the convergence or divergence of the series  using any appropriate test. ​

Accepted Answer

The answer of Determine the convergence or divergence of the...

Question 19

Identify the most appropriate test to be used to determine whether the series  converges or diverges. ​

Accepted Answer

The answer of Identify the most appropriate test to be...

Question 20

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

Accepted Answer

The answer of Determine whether the series  converges conditionally...

Quiz 9: Infinite Series

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

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

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

True or false: The series converges.

Determine the convergence or divergence of the series using any appropriate test.

The terms of a series are defined recursively.Determine the convergence or divergence of the series.Explain your reasoning. ,

Determine the convergence or divergence of the series using any appropriate test.

Identify the most appropriate test to be used to determine whether the series converges or diverges.

True or false: The series diverges .

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

Identify the most appropriate test to be used to determine whether the series converges or diverges.

True or false: The series diverges.

Find the values of x for which the series converges.

Find the values of x for which the series converges.

True or false: The series diverges.

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

Determine the convergence or divergence of the series using any appropriate test.

Identify the most appropriate test to be used to determine whether the series converges or diverges.

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

Quiz 9: Infinite Series

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​ ​

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​ ​

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

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

True or false: The series converges. ​

Determine the convergence or divergence of the series using any appropriate test. ​

The terms of a series are defined recursively.Determine the convergence or divergence of the series.Explain your reasoning. ​ , ​

Determine the convergence or divergence of the series using any appropriate test. ​

Identify the most appropriate test to be used to determine whether the series converges or diverges. ​

True or false: The series diverges . ​

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

Identify the most appropriate test to be used to determine whether the series converges or diverges.

True or false: The series diverges. ​

Find the values of x for which the series converges.

Find the values of x for which the series converges. ​

True or false: The series diverges. ​

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used. ​ ​

Determine the convergence or divergence of the series using any appropriate test. ​

Identify the most appropriate test to be used to determine whether the series converges or diverges. ​

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

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

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

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

True or false: The series converges.

Determine the convergence or divergence of the series using any appropriate test.

The terms of a series are defined recursively.Determine the convergence or divergence of the series.Explain your reasoning. ,

Determine the convergence or divergence of the series using any appropriate test.

Identify the most appropriate test to be used to determine whether the series converges or diverges.

True or false: The series diverges .

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

True or false: The series diverges.

Find the values of x for which the series converges.

True or false: The series diverges.

Determine the convergence or divergence of the series using any appropriate test from this chapter.Identify the test used.

Determine the convergence or divergence of the series using any appropriate test.

Identify the most appropriate test to be used to determine whether the series converges or diverges.

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