
Exercises relating to alternating series and absolute or conditional convergence.

For the infinite series the function $1/\ln n$ isis not positive and isis not decreasing when $n \geq 2$. Furthermore it doesdoes not tend to zero as $n \rightarrow \infty$. The alternating series test doesdoes not apply and implies convergenceimplies divergencesays nothing about the series .
For the infinite series the function $1/(\ln n+(-1)^n \sqrt {\ln n})$ isis not positive and isis not decreasing when $n \geq 2$. Furthermore it doesdoes not tend to zero as $n \rightarrow \infty$. The alternating series test doesdoes not apply and implies convergenceimplies divergencesays nothing about the series .
For the infinite series the function $1/(\ln n - (\ln n)^{-1/2})$ isis not positive and isis not decreasing when $n \geq 3$. Furthermore it doesdoes not tend to zero as $n \rightarrow \infty$. The alternating series test doesdoes not apply and implies convergenceimplies divergencesays nothing about the series .
For the infinite series the function $1/(\ln n+(-1)^n \sqrt {\ln n})$ isis not positive and isis not decreasing when $n \geq 2$. Furthermore it doesdoes not tend to zero as $n \rightarrow \infty$. The alternating series test doesdoes not apply and implies convergenceimplies divergencesays nothing about the series .
For the infinite series the function $1/(e^n)$ isis not positive and isis not decreasing when $n \geq 1$. Furthermore it doesdoes not tend to zero as $n \rightarrow \infty$. The alternating series test doesdoes not apply and implies convergenceimplies divergencesays nothing about the series .
First determine whether the series is alternating.
Does the alternating series test apply to the series?
Yes No, it’s not alternating No, the terms are not decreasing No, the terms do not go to zero
If yes, for what minimum value of $N$ can you be certain that differs from the sum of the series by at most $10^{-3}$? If no such $N$ exists, write N/A.
If the alternating series test applies, we would need the magnitude (i.e., absolute value) of the first term not included in the partial sum to be no greater than $10^{-3}$.
Find an interval of length $\frac {1}{4}$ which contains the sum of the infinite series
Partial sums of an alternating series also alternate above and below the sum of the series itself.

### Sample Quiz Questions

For each series below, determine whether it converges absolutely (A), converges conditionally (C), or diverges (D). Show how you used convergence tests to arrive at your answer.

I: C, II: D, III: D I: C, II: A, III: C I: A, II: C, III: A I: D, II: C, III: D I: C, II: D, III: C I: C, II: A, III: A

For each series below, determine whether it converges absolutely (A), converges conditionally (C), or diverges (D). Show how you used convergence tests to arrive at your answer.

I: D, II: D, III: D I: D, II: A, III: C I: C, II: C, III: A I: A, II: C, III: D I: D, II: D, III: C I: D, II: A, III: A

Which of the following intervals contains the value of the infinite series

$\displaystyle \left [ \frac {1}{4}, \frac {1}{3} \right ]$ $\displaystyle \left [ \frac {1}{3}, \frac {1}{2} \right ]$ $\displaystyle \left [ \frac {1}{2}, \frac {7}{12} \right ]$ $\displaystyle \left [ \frac {7}{12}, \frac {5}{6} \right ]$ $\displaystyle \left [ \frac {5}{6}, \frac {11}{12} \right ]$ $\displaystyle \left [ \frac {11}{12}, \frac {7}{6} \right ]$

### Sample Exam Questions

Determine whether the following series converge absolutely (A), converge conditionally (C), or diverge (D). For full credit be sure to explain your reasoning and specify which tests were used.

both A one A, the other C one A, the other D both C one C, the other D both D