Q: Which of the following is a necessary condition for the convergence of the infinite series $$\sum_{n=m}^{\infty} a_n$$?

$$\lim_{n \to \infty} a_n = 0$$

Q: Which of the following statements about the root test is correct?

If $$\limsup_{n \to \infty} |a_n|^{1/n} \u003c 1$$, the series converges absolutely.

Question 1

Which of the following statements about finite series is TRUE?

Accepted Answer

The sum of a finite series is independent of the order of the terms.

Question 2

Given the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$, which of the following is TRUE?

Accepted Answer

The series converges because it is a $$p-$$series with $$p = 2 \u003e 1.$$

Question 3

Suppose $$a_n = (−1)^n \frac{1}{n}. $$Which test can be used to show that $$\sum_{n=1}^{\infty} a_n$$ converges?

Accepted Answer

Alternating series test

Question 4

Which of the following is a necessary condition for the convergence of the infinite series $$\sum_{n=m}^{\infty} a_n$$?

Accepted Answer

$$\lim_{n 	o \infty} a_n = 0$$

Question 5

Let $$\sum_{n=0}^{\infty} x^n be a $$geometric series. For which values of $$x$$ does the series converge?

Accepted Answer

$$|x| \u003c 1$$

Question 6

If $$\sum_{n=1}^{\infty} a_n$$ converges absolutely, what can be said about any rearrangement of its terms?

Accepted Answer

The rearranged series converges to the same sum.

Question 7

Which of the following series is divergent?

Accepted Answer

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

Question 8

Suppose $$\sum_{n=1}^{\infty} a_n is a $$series of non-negative terms. Which of the following is equivalent to its convergence?

Accepted Answer

The sequence of partial sums is bounded above.

Question 9

Which of the following statements about the root test is correct?

Accepted Answer

If $$\limsup_{n 	o \infty} |a_n|^{1/n} \u003c 1$$, the series converges absolutely.

Question 10

A medical researcher is analyzing the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n$$^{1.5}$$} to $$model the decay of a drug in the bloodstream. What can be concluded?

Accepted Answer

The series converges because $$1.5 \u003e 1 (by $$the $$p-$$series test).

Question 11

Which of the following is a correct application of the comparison test?

Accepted Answer

If $$0 \leq a_n \leq b_n$$ for all $$n$$ and $$\sum b_n$$ converges, then $$\sum a_n$$ converges.

Question 12

Which of the following series is only conditionally convergent?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$

Question 13

A chemist is using the ratio test to determine the convergence of $$\sum_{n=1}^{\infty} \frac{n!}{2^n}. $$What is the result?

Accepted Answer

The series diverges because the ratio test yields a limit greater than 1.

Question 14

Which of the following is a telescoping series, and what is its sum if $$a_n 	o 0$$?

Accepted Answer

$$\sum_{n=0}^{\infty} (a_n - a_{n+1})$$ with sum $$a_0$$

Question 15

If $$\sum_{n=1}^{\infty} a_n is $$conditionally convergent, what can happen if the terms are rearranged?

Accepted Answer

The sum can change to any real number or even diverge.

Question 16

Which of the following is the correct statement of the absolute convergence test?

Accepted Answer

If $$\sum |a_n|$$ converges, then $$\sum a_n$$ converges.

Question 17

A financial analyst is modeling compound interest using the series $$\sum_{n=0}^{\infty} r^n$$ with $$|r| \u003c 1. $$What is the sum of this series?

Accepted Answer

$$\frac{1}{1 - r}$$

Question 18

Which of the following is a correct use of the index shift property for series?

Accepted Answer

$$\sum_{n=m}^{\infty} a_n = \sum_{n=m+k}^{\infty} a_n + \sum_{n=m}^{m+k-1} a_n$$

Series and Convergence: Finite and Infinite Series in Precalculus

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