Question 1

Write the first five terms of the sequence of partial sums. ​  ​

Accepted Answer

The answer of Write the first five terms of the...

Question 2

Determine the convergence or divergence of the series. ​  ​

Accepted Answer

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

Question 3

Sketch the graph of the sequence of partial sum of the series  . ​

Accepted Answer

The partial sum of the given series is as follows:
S1 = 1
S2 = 1 + 4
S3 = 1 + 4 + 16
S4 = 1 + 4 + 16 + 64
S5 = 1 + 4 + 16 + 64 + 256
...
We can see that each partial sum is obtained by adding the square of the previous term to the current term. Therefore, the correct choice must have the correct sequence of squares.
Option C is the only one that has the correct sequence of squares (1, 4, 16, 64, 256, ...) and therefore the correct partial sums.

Question 4

Write the first five terms of the sequence of partial sums. ​  ​

Accepted Answer

The answer of Write the first five terms of the...

Question 5

Find the sum of the convergent series  ​

Accepted Answer

The answer of Find the sum of the convergent series...

Question 6

True or false.The series  is convergent.
​

Accepted Answer

The answer of True or false.The series  is convergent.
​...

Question 7

Determine the convergence or divergence of the series. ​  ​

Accepted Answer

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

Question 8

Find the sum of the convergent series. ​  ​

Accepted Answer

The answer of Find the sum of the convergent series....

Question 9

Find the sum of the convergent series  . ​

Accepted Answer

The series is in the form of a geometric series with ratio e. Thus, the sum of the series is:
$\frac{11}{1-e} = \frac{11}{1-2.718281828\dots} \approx \frac{11}{0.6382} \approx 17.2381$
Therefore, the answer is B) 11eac44f_ec46_9337_bedf_970ff7284593_TB7497_11.

Question 10

Determine the convergence or divergence of the series. ​  ​

Accepted Answer

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

Question 11

Write the repeating decimal  as a geometric series. ​

Accepted Answer

To convert the repeating decimal to a geometric series, we need to first write it in the form:
a + ar + ar^2 + ar^3 + ...

where a is the first term and r is the common ratio.

Starting from the end of the repeating block and working backwards, we can see that the last group of digits before the "TB7497_11" is "1af7". This can be written as:
1af7 / 10^4

Moving to the left, the next group of digits is "83f9b729". We can add this to the previous fraction by multiplying it by the common ratio, which is 1/10^8 since there are 8 digits between the two groups:
(1af7 / 10^4) + (83f9b729 / 10^12)

Continuing this process, we can add the following groups of digits to the previous sum by multiplying by the common ratio:
_e15b_bedf (common ratio = 1/10^4)
_ec46 (common ratio = 1/10^4)
_11 (common ratio = 1/10^2 since there are 2 digits between this group and the previous)

Putting this all together, we get:
(1af7 / 10^4) + (83f9b729 / 10^12) + (_e15b_bedf / 10^16) + (_ec46 / 10^20) + (_11 / 10^22) + ...

This can be simplified as:
(1af7 / 10^4) / (1 - 1/10^4) = 1af7 / 9999 * 10^4
+ (83f9b729 / 10^12) / (1 - 1/10^4) = 83f9b729 / 9999 * 10^8
+ (_e15b_bedf / 10^16) / (1 - 1/10^4) = e15b_bedf / 9999 * 10^12
+ (_ec46 / 10^20) / (1 - 1/10^4) = ec46 / 9999 * 10^16
+ (_11 / 10^22) / (1 - 1/10^2) = 11 / 99 * 10^22
+ ...

Therefore, the repeating decimal 11eac44f_ec46_e15b_bedf_83f9b7291af7_TB7497_11 can be written as the geometric series:
(1af7 / 9999 * 10^4) + (83f9b729 / 9999 * 10^8) + (e15b_bedf / 9999 * 10^12) + (ec46 / 9999 * 10^16) + (11 / 99 * 10^22) + ...

Question 12

Find the positive values of p for which the series  converges. ​

Accepted Answer

The given series can be written in the form of general term as 
$a_n= \frac{p^n}{n!}$
Now, using the ratio test, 
$\frac{a_{n+1}}{a_n}=\frac{p}{n+1}$
For the series to converge, the above ratio should be less than 1. Therefore, 
$\frac{p}{n+1}<1$
Solving the above inequality, we get
$p<n+1$
As the series starts from n=0, we can replace n with n+1 in the above inequality to get 
$p<(n+1)+1=n+2$
Therefore, the series converges for values of p less than n+2 for any n. Since p has to be positive, we can take the smallest value of n i.e. n=0. So, p<2. 
Hence, the series converges only for values of p less than 2. 
Out of the given answer options, only option D satisfies this condition.

Question 13

Write the first five terms of the sequence of partial sums. ​  ​

Accepted Answer

The answer of Write the first five terms of the...

Question 14

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

Accepted Answer

The answer of Find all values of x for which...

Question 15

Find the sum of the convergent series. ​  ​

Accepted Answer

The answer of Find the sum of the convergent series....

Question 16

True or false.The series  is divergent.
​

Accepted Answer

We cannot determine if the series is divergent or not without more information about the terms of the series.

Question 17

True or false.The infinite series  .
​

Accepted Answer

The given series is not well-defined as the term after the underscore is not specified. Therefore, it is neither true nor false.

Question 18

Determine the convergence or divergence of the series. ​  ​

Accepted Answer

The series does not approach zero as n approaches infinity, and therefore diverges.

Question 19

Find the positive values of p for which the series  converges. ​

Accepted Answer

The answer of Find the positive values of p for...

Question 20

Find the sum of the convergent series. ​  ​

Accepted Answer

The answer of Find the sum of the convergent series....

Quiz 9: Infinite Series

Write the first five terms of the sequence of partial sums. ​ ​

Determine the convergence or divergence of the series. ​ ​

Sketch the graph of the sequence of partial sum of the series . ​

Write the first five terms of the sequence of partial sums. ​ ​

Find the sum of the convergent series ​

True or false.The series is convergent. ​

Determine the convergence or divergence of the series. ​ ​

Find the sum of the convergent series. ​ ​

Find the sum of the convergent series . ​

Determine the convergence or divergence of the series. ​ ​

Write the repeating decimal as a geometric series. ​

Find the positive values of p for which the series converges. ​

Write the first five terms of the sequence of partial sums. ​ ​

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

Find the sum of the convergent series. ​ ​

True or false.The series is divergent. ​

True or false.The infinite series . ​

Determine the convergence or divergence of the series. ​ ​

Find the positive values of p for which the series converges. ​

Find the sum of the convergent series. ​ ​

Write the first five terms of the sequence of partial sums.

Determine the convergence or divergence of the series.

Sketch the graph of the sequence of partial sum of the series .

Write the first five terms of the sequence of partial sums.

Find the sum of the convergent series

True or false.The series is convergent.

Determine the convergence or divergence of the series.

Find the sum of the convergent series.

Find the sum of the convergent series .

Determine the convergence or divergence of the series.

Write the repeating decimal as a geometric series.

Find the positive values of p for which the series converges.

Write the first five terms of the sequence of partial sums.

Find all values of x for which the series converges.

Find the sum of the convergent series.

True or false.The series is divergent.

True or false.The infinite series .

Determine the convergence or divergence of the series.

Find the positive values of p for which the series converges.

Find the sum of the convergent series.