Question 1

Use the Limit Comparison Test to determine the convergence or divergence of the series  . ​

Accepted Answer

The answer of Use the Limit Comparison Test to determine...

Question 2

True or false: The series  converges.
​

Accepted Answer

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

Question 3

Use the Direct Comparison Test to determine the convergence or divergence of the series  . ​

Accepted Answer

The answer of Use the Direct Comparison Test to determine...

Question 4

Use the Integral Test to determine the convergence or divergence of the series. ​  ​

Accepted Answer

We need to use the Integral Test, which states that if f(x) is a positive, continuous, and decreasing function for x ≥ k and f(n) = an for all n ≥ k, then the series ∑an converges if and only if the improper integral  ∫k to infinity f(x) dx converges.

Let f(x) = e^(-x^2) / x. Then, f(x) is a positive, continuous, and decreasing function for x ≥ 1. Also, f(n) = e^(-n^2) / n, which is the nth term of the given series.

Now, we need to evaluate the improper integral ∫1 to infinity e^(-x^2) / x dx. This integral cannot be evaluated using elementary methods, so we need to use a calculus technique called substitution. Let u = x^2, then du/dx = 2x and dx = du / (2x). The integral becomes ∫1 to infinity e^(-u) / (2u)^0.5 du.

Using the substitution u = y^2, we get the integral as  ∫1 to infinity e^(-y^2) dy. This integral is called the Gaussian integral and has been evaluated (with the help of complex analysis) to be ∫-infinity to infinity e^(-x^2) dx = (pi)^0.5. It follows that ∫1 to infinity e^(-x^2) / x dx converges.

Therefore, by the Integral Test, the series ∑e^(-n^2) / n converges.

Question 5

True or false: The series  converges.
​

Accepted Answer

It is not possible to determine the convergence of a series just by its expression. More information is needed, such as the terms of the series and their behavior.

Question 6

Use the Integral Test to determine the convergence or divergence of the series. ​  ​

Accepted Answer

The answer of Use the Integral Test to determine the...

Question 7

Use Theorem 9.11 to determine the convergence or divergence of the series. ​  ​

Accepted Answer

The answer of Use Theorem 9.11 to determine the convergence...

Question 8

Determine the convergence or divergence of the series  . ​

Accepted Answer

The series diverges by the divergence test because the general term does not approach zero as n approaches infinity.

Question 9

Use the Integral Test to determine the convergence or divergence of the series. ​  ​

Accepted Answer

Consider the function f(x) = ln(x) / x. The function is defined for all x > 0 and is continuous and positive on the interval [1, ∞).

Now, for n ≥ 2, we have

f(n) = ln(n) / n ≤ ln(n) / 2

This is true because 2 ≤ n; since ln(x) is an increasing function, it follows that ln(n) ≥ ln(2) ≈ 0.69.

Using this inequality, we can estimate the nth term of the series:

a_n = (ln(n))^2 / n^2 
  = f(n) / n 
  ≤ (ln(n) / 2) / n 
  = ln(n) / (2n)

Integrating both sides from 1 to infinity, we get:

∫(1 to ∞) a_n dn ≤ ∫(1 to ∞) ln(x) / (2x) dx

The integral on the right-hand side can be evaluated using integration by parts:

u = ln(x), dv = dx / (2x) 
du = dx / x, v = (1/2) ln(x)

∫(1 to ∞) ln(x) / (2x) dx 
= [(1/2) ln(x)^2]_(1 to ∞) - ∫(1 to ∞) dx / (2x) ln(x) 
= (1/2) lim_(n→∞) ln(n)^2 - [(1/2) ln(x)]_(1 to ∞) 
= ∞

Since the improper integral diverges, it follows that the series ∑ a_n also diverges by the Integral Test. Therefore, the answer is A) diverges.

Question 10

True or false: The series  converges.
​

Accepted Answer

This is not a well-defined series. It appears to be a random collection of characters and numbers. A series must have a defined pattern or formula to determine whether it converges or diverges.

Question 11

True or false: The series  converges.
​

Accepted Answer

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

Question 12

Use Theorem 9.11 to determine the convergence or divergence of the series. ​  ​

Accepted Answer

The series does not satisfy the conditions of Theorem 9.11, as the nth term does not approach zero as n approaches infinity. Therefore, we cannot use this theorem to determine the convergence or divergence of the series. Additional tests, such as the divergence test or comparison test, may be necessary.

Question 13

True or false: The series  diverges.
​

Accepted Answer

As the series does not have a pattern and does not converge, it diverges.

Question 14

Determine the convergence or divergence of the series  . ​

Accepted Answer

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

Question 15

Use the polynomial test to determine whether the series  converges or diverges. ​

Accepted Answer

The answer of Use the polynomial test to determine whether...

Question 16

Consider the series  .The sum of the series is  Find the sum of the series  . ​

Accepted Answer

The answer of Consider the series  .The sum of...

Question 17

Use the Integral Test to determine the convergence or divergence of the series. ​  ​

Accepted Answer

We use the integral test to determine convergence or divergence of the series. Let f(x) = 1/(x^2+1). Since f(x) is continuous, positive and decreasing for x≥1, then by the integral test, we get:

∫₁ to infinity 1/(x^2+1) dx ≤ ∑_(n=1)^∞ 1/(n^2+1)

Using the substitution u = x/2, we rewrite the integral as

(1/2) ∫_(1/2) to infinity 1/(u^2+1) du

and evaluating the integral, we get:

(1/2)[tan^-1(u)]_(1/2) to infinity = (1/2)[π/2- tan^-1(1/2)]

Since the integral is convergent, and the series is greater than this integral, then the series must also be divergent. Therefore, the correct choice is A) diverges.

Question 18

Use Theorem 9.11 to determine the convergence or divergence of the series. ​  ​

Accepted Answer

Theorem 9.11 states that if the limit of the absolute value of the terms of a series is zero, then the series converges. In this case, we can use the limit comparison test with the series 1/n^3.

We have:

lim (n → ∞) (11/n^3) / (1/n^3) 
= lim (n → ∞) 11 
= 11

Since this limit is finite and positive, we conclude that the series 11/n^3 converges if and only if the series 1/n^3 converges. But the series 1/n^3 is a known convergent p-series with p = 3 > 1.

Therefore, by the limit comparison test, we conclude that the series 11/n^3 also converges. Thus, the answer is B.

Question 19

Use the polynomial test to determine whether the series  converges or diverges. ​

Accepted Answer

The answer of Use the polynomial test to determine whether...

Question 20

Use the Limit Comparison Test to determine the convergence or divergence of the series  . ​

Accepted Answer

The answer of Use the Limit Comparison Test to determine...

Quiz 9: Infinite Series

Use the Limit Comparison Test to determine the convergence or divergence of the series . ​

True or false: The series converges. ​

Use the Direct Comparison Test to determine the convergence or divergence of the series . ​

Use the Integral Test to determine the convergence or divergence of the series. ​ ​

True or false: The series converges. ​

Use the Integral Test to determine the convergence or divergence of the series. ​ ​

Use Theorem 9.11 to determine the convergence or divergence of the series. ​ ​

Determine the convergence or divergence of the series . ​

Use the Integral Test to determine the convergence or divergence of the series. ​ ​

True or false: The series converges. ​

True or false: The series converges. ​

Use Theorem 9.11 to determine the convergence or divergence of the series. ​ ​

True or false: The series diverges. ​

Determine the convergence or divergence of the series . ​

Use the polynomial test to determine whether the series converges or diverges. ​

Consider the series .The sum of the series is Find the sum of the series . ​

Use the Integral Test to determine the convergence or divergence of the series. ​ ​

Use Theorem 9.11 to determine the convergence or divergence of the series. ​ ​

Use the polynomial test to determine whether the series converges or diverges. ​

Use the Limit Comparison Test to determine the convergence or divergence of the series . ​

Use the Limit Comparison Test to determine the convergence or divergence of the series .

True or false: The series converges.

Use the Direct Comparison Test to determine the convergence or divergence of the series .

Use the Integral Test to determine the convergence or divergence of the series.

True or false: The series converges.

Use the Integral Test to determine the convergence or divergence of the series.

Use Theorem 9.11 to determine the convergence or divergence of the series.

Determine the convergence or divergence of the series .

Use the Integral Test to determine the convergence or divergence of the series.

True or false: The series converges.

True or false: The series converges.

Use Theorem 9.11 to determine the convergence or divergence of the series.

True or false: The series diverges.

Determine the convergence or divergence of the series .

Use the polynomial test to determine whether the series converges or diverges.

Consider the series .The sum of the series is Find the sum of the series .

Use the Integral Test to determine the convergence or divergence of the series.

Use Theorem 9.11 to determine the convergence or divergence of the series.

Use the polynomial test to determine whether the series converges or diverges.

Use the Limit Comparison Test to determine the convergence or divergence of the series .