Solution verification of the convergence of $ \sum_{n=1}^\infty \frac{(-1)^n}{x+n} $

Question

This is an exam problem from the analysis 1 course I attend.

Problem: Test the following series for absolute and conditional convergence $$ \sum_{n=1}^\infty \frac{(-1)^n}{x+n} $$ on the interval $(0,+\infty)$.

Solution:

Testing absokute convergence:

$$\sum_{n=1}^\infty \left| \frac{(-1)^n}{x+n}\right|=\sum_{n=1}^\infty \frac{1}{x+n} $$

1st way to test for absolute convergence using the Raabe criteria:

$$\lim_{n \to \infty}\frac{x+n+1}{x+n}-1=0\ \ \text{,}$$

for all $x \in (0,+\infty)$.

Since $0 < 1$, according to Raabe criteria the series absolutely diverge.

2nd way to test for absolute convergence using the comparison test:

$\text{Since}\ \ \frac {1}{n} < \frac{1}{x+n}\ \ \text{for all }x \in (0,+\infty),\ \text{and since the Harmonic series diverge, whose general term is}\ \ \frac{1}{n},\text {the series absolutely diverge according to the comparison test.}$

Testing conditional convergence(using Leibniz convergence criteria)

1. $$\left|a_n+1\right| < \left|a_n\right|, \frac{1}{x+n+1} < \frac{1}{x+n} $$ We can see that the sequence is monotonically decreasing for all $x \in (0,+\infty)$.

2. $$ \lim_{n \to \infty} \frac{1}{x+n}=0\ \ \text{,}$$

for all $x \in (0,+\infty)$.

And thereby, by the Leibniz criteria, the series converge conditionally.

Is everything alright with the solution ? Feel free to add suggestions if you have any. Thanks.

Answer 1

Yes, the Leibniz test shows the conditional convergence. However, your use of the comparison test is wrong since you start with,

$$\frac{1}{n} < \frac{1}{n+x}$$

which is not true for $x\in(0,\infty)$. Think of $n=1$ with $x=1$,

$$\frac{1}{n} = 1 > \frac{1}{2} = \frac{1}{n+x}$$

You need to fix $x\in(0,\infty)$ and then lower bound $\frac{1}{x+n}$ by a term from the harmonic series, but $\frac{1}{n}$ is not the one you want.

Answer 2

Unfortunately, your use of the Raabe criterion was also incorrect. The Raabe test involves the computation of the limit $$ \lim_{n\to\infty} n \left(\frac{a_n}{a_{n+1}} - 1\right) = \lim_{n\to\infty} n \left(\frac{x+n+1}{x+n} - 1\right), $$ but in your answer, you dropped the factor of $n$. With this corrected, the Raabe criterion is inconclusive: $$ \lim_{n\to\infty} n \left(\frac{x+n+1}{x+n} - 1\right) = \lim_{n\to\infty} \frac{n}{x+n}= 1. $$