Determine the convergence of $\sum_{n=2}^{\infty} \frac{(-1)^n}{(-1)^n+n}$

Question

I've been really stuck on this infinite series.

$\sum_{n=2}^{\infty} \frac{(-1)^n}{(-1)^n+n}$

Trying to determine if it converges or diverges.

As far as I've got is seperating it into two partial sums (odds and evens) but even this doesn't help me.

Any ideas?

Thanks

Answer 1

The key word is Leibnitz. But there is a small but, the sequences of the absolute values for which we build the series is not decreasing to zero. One can take care, and still apply Leibnitz... But ok, this is a special problem with a possible special solution, it is always good to "see the series", then maybe even more as a bonus, how quick/slow it converges/diverges...: $$ \begin{aligned} \sum_{n\ge 2}\frac{(-1)^n}{n+(-1)^n} &= \lim_{N\to\infty} \underbrace{\sum_{2\le n\le N}\frac{(-1)^n}{n+(-1)^n}}_{\text{Notation: }a_N} \\ &\qquad\text{ so let us consider an odd index general term $a_{2N+1}$, $N\ge 1$} \\ a_{2N+1} &= \left(\frac 13-\frac 12\right) + \left(\frac 15-\frac 14\right) + \left(\frac 17-\frac 16\right) + \dots + \left(\frac 1{2N+1}-\frac 1{2N}\right) \\ &= -\frac 1{2\cdot 3} -\frac 1{4\cdot 5} -\frac 1{6\cdot 7} -\dots- \frac 1{2N(2N+1)}\ . \end{aligned} $$ It is enough to consider the convergence / divergence of this subsequence, because the subsequence of even indexed terms in $(a_N)$ differs from this one by a zero sequence.

Now $(a_{2N+1})$ has negative terms, and is strictly decreasing. We show it is bounded (below), and thus the convergence. I prefer positive numbers, for instance for typographical reasons, so consider $(-a_{2N+1})$, and let us bound it as follows: $$ \begin{aligned} &\frac 1{2\cdot 3} +\frac 1{4\cdot 5} +\dots+ \frac 1{2N(2N+1)} \\ &\qquad\le \frac 1{1\cdot 3} +\frac 1{3\cdot 5} +\dots+ \frac 1{(2N-1)(2N+1)} \\ &\qquad= \frac 12\left(\frac 11-\frac 13\right) + \frac 12\left(\frac 13-\frac 15\right) + \dots + \frac 12\left(\frac 1{2N-1}-\frac 1{2+N}\right) \\ &\qquad=\frac 12\left(\frac 11-\frac 1{2+N}\right) \\ &\qquad\longrightarrow \frac 12 \ . \end{aligned} $$ So we expect that the given sequence is converging to a real number $\ge -\frac 12$.

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Note: It is always important in such cases to see the numerical value, pari/gp gives

? sum(n=2, 2000001, 1.0*(-1)^n/(n+(-1)^n) )
%8 = -0.30685256944024219045776794885429218552
? sum(n=1, 1000000, 1.0/(2*n)/(2*n+1) )
%9 = 0.30685256944024219045776794885429144764

Answer 2

Looking at the partial sums,

$\begin{array}\\ \sum_{n=2}^{2m+1} \frac{(-1)^n}{(-1)^n+n} &=\sum_{n=1}^{m} \left(\frac{1}{2n+1}-\frac{1}{2n}\right) \qquad 2n \text{ and } 2n+1\\ &=\sum_{n=1}^{m} -\frac{1}{2n(2n+1)}\\ \end{array} $

and this sum converges absolutely by comparison with $\sum \frac1{n^2}$.

To see this in an elementary way, starting at $n=2$ which does not affect convergence,

$\begin{array}\\ \sum_{n=2}^{m} \frac{1}{2n(2n+1)} &\lt \sum_{n=2}^{m} \frac{1}{2n(2n-2)}\\ &= \frac14\sum_{n=2}^{m} \frac{1}{n(n-1)}\\ &= \frac14\sum_{n=2}^{m} (\frac{1}{n-1}-\frac1{n})\\ &= \frac14(1-\frac1{m})\\ &\lt \frac14\\ \end{array} $

so, restoring $=1$,

$\sum_{n=1}^{m} \frac{1}{2n(2n+1)} \lt \frac16+\frac14 =\frac13 $.

Note: Wolfy says the sum is $1-\ln(2) \approx 0.306852819440054690582767878541823431924499865639744745879... $

Answer 3

Let's take a look at the 2 series:

(1) $\sum_{n=2}^{\infty}-\frac{1}{n^2+(-1)^nn}$

(2) $\sum_{n=2}^{\infty}\frac{(-1)^n}{n}$

Series (1) converges absolutely (comparing with $\sum\frac{1}{n^2}$).

Series (2) is the Leibniz Series and converges conditionally.

Using limit laws, the series obtained by summing (1) and (2) converges as well:

$\sum_{n=2}^{\infty}\frac{(-1)^n}{n}+\sum_{n=2}^{\infty}-\frac{1}{n^2+(-1)^nn}=\sum_{n=2}^{\infty}\frac{n(-1)^n+(-1)^{2}}{n}=\sum_{n=2}^{\infty}\frac{(-1)^n}{(-1)^n+n}$

Answer 4

We have

$$\frac{(-1)^n}{(-1)^n+n}=\frac{(-1)^n}{n}\left(1+\frac{(-1)}{n}\right)^{-1}=\frac{(-1)^n}{n}\left(1+O\left(\frac1n\right)\right)=\frac{(-1)^n}{n}+O\left(\frac1{n^2}\right)$$

• The series $\sum \frac{(-1)^n}n$ converges by Leibniz criterion
• the series $\sum O\left(\frac1{n^2}\right)$ is convergent by comparison with Riemann series

hence the given series is also convergent.

Answer 5

Note that

$$\sum_{n=2}^m \frac{(-1)^n}{(-1)^n + n} = \sum_{n=2}^m \frac{(-1)^n}{(-1)^n + n}\frac{n - (-1)^n}{n - (-1)^n} \\= \underbrace{\sum_{n=2}^m \frac{(-1)^nn}{n^2 - 1}}_{\text{(1)}} - \underbrace{\sum_{n=2}^m \frac{1}{n^2 -1 }}_{\text{(2)}}, $$

where (1) is a convergent alternating series and (2) converges since $\frac{1}{n^2-1} \sim \frac{1}{n^2}$