Q: Which statement about the sum of a series and the value of the corresponding improper integral is correct?

In general, $$\sum_{n=1}^{\infty} a_n \neq \int_{1}^{\infty} f(x) dx$$

Question 1

Which of the following statements best describes the Integral Test for series convergence?

Accepted Answer

If $$f(x) is $$continuous, positive, and decreasing for $$x \geq N$$, then $$\sum_{n=N}^{\infty} a_n$$ converges if and only if $$\int_{N}^{\infty} f(x) dx$$ converges.

Question 2

For which values of $$p$$ does the series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ converge?

Accepted Answer

For $$p \u003e 1$$

Question 3

Consider the series $$\sum_{n=1}^{\infty} \frac{\ln n}{n}. $$Which test can be used to determine its convergence, and what is the result?

Accepted Answer

Integral Test; the series diverges.

Question 4

Which of the following is a correct remainder estimate for the sum $$S of a $$convergent series $$\sum_{n=1}^{\infty} a_n$$ using the Integral Test?

Accepted Answer

$$\int_{n+1}^{\infty} f(x) dx \leq R_n \leq \int_{n}^{\infty} f(x) dx$$

Question 5

Suppose you want to estimate the sum $$\sum_{n=1}^{\infty} \frac{1}{n^3$} to $$within $$0.0005. $$How many terms are required?

Accepted Answer

32 terms

Question 6

Which of the following series is divergent?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

Question 7

Given the series $$\sum_{n=1}^{\infty} \frac{1}{n$$^{1/2}$$}$$, what does the table of partial sums suggest about its convergence?

Accepted Answer

The partial sums increase without bound, so the series diverges.

Question 8

Which statement about the sum of a series and the value of the corresponding improper integral is correct?

Accepted Answer

In general, $$\sum_{n=1}^{\infty} a_n 
eq \int_{1}^{\infty} f(x) dx$$

Question 9

Which of the following is a correct application of the Direct Comparison Test?

Accepted Answer

If $$0 \leq a_n \leq b_n$$ and $$\sum b_n$$ converges, then $$\sum a_n$$ also converges.

Question 10

For the series $$\sum_{n=1}^{\infty} \frac{5}{2n^2$}$$, which test and result apply?

Accepted Answer

It is a constant times a $$p-$$series with $$p=2\u003e1$$, so it converges.

Question 11

Suppose $$a_n = \frac{1}{2n-1}$$ and $$b_n = \left(\frac{1}{2}\right)^n. $$Using the Limit Comparison Test, what is the result?

Accepted Answer

The limit is positive and finite, and $$\sum b_n$$ converges, so $$\sum a_n$$ converges.

Question 12

Which of the following is NOT a requirement for applying the Integral Test?

Accepted Answer

The function $$f(x)$$ must be always decreasing for all $$x \geq N$$

Question 13

If $$\sum_{n=1}^{\infty} \frac{1}{n^p} is $$convergent, what can be said about the remainder $$R_n$$ after $$n$$ terms?

Accepted Answer

$$R_n \leq \int_{n}^{\infty} \frac{1}{x^p} dx$$

Question 14

Which of the following statements about the Limit Comparison Test is correct?

Accepted Answer

If $$\lim_{n 	o \infty} \frac{a_n}{b_n} = c \u003e 0$$, then $$\sum a_n$$ and $$\sum b_n$$ both converge or both diverge.

Question 15

Estimate the error involved in approximating $$\sum_{n=1}^{\infty} \frac{1}{n^3$} by $$the sum of the first 10 terms.

Accepted Answer

The error is at most $$0.005$$

Question 16

Which of the following series is convergent by the Comparison Test?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{1}{n^3$+1}$$

Question 17

If $$\sum_{n=1}^{\infty} b_n is a $$geometric series with $$|r| \u003c 1$$, and $$a_n \leq b_n$$ for all $$n$$, what can be said about $$\sum_{n=1}^{\infty} a_n$$?

Accepted Answer

$$\sum_{n=1}^{\infty} a_n$$ converges

Question 18

Which of the following is a correct statement about estimating the sum of a convergent series using bounds from the Integral Test?

Accepted Answer

The sum $$s$$ satisfies $$s_n + \int_{n+1}^{\infty} f(x) dx \leq s \leq s_n + \int_{n}^{\infty} f(x) dx$$

Integral and Comparison Tests for Series Convergence

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