Alternating series test [Question of monotone decreasing sequence]

Question

I have to show that

Let $n\in\mathbb{N}\cup\{0\}$ and let $(a_n)_{n}$ be defined as $a_n=(-1)^n\frac{n}{n^2+1}$. Check convergence of the series $\sum_{n=0}^{\infty} a_n$.

Using the alternating series test, the series is convergent if $|a_n|$ is a monotone decreasing sequence and $\lim_{n\to\infty}|a_n|=0$.

Using induction to prove that $|a_n|$ is a monotone decreasing sequence, $P(1)$ would have me prove that $|a_0|\ge|a_1|$ since $n\in\mathbb{N}\cup\{0\}$, which is obviously not true: $$ |a_0|=1\cdot 0=0\not\ge |a_1|=|(-1)\cdot \frac{1}{2}|=\frac{1}{2} $$ Though in the next question it is assumed that you have found that the series is convergent with the alternating series test, which would be true if $n\in\mathbb{N}$.

Is my induction hypothesis wrong?

Answer 1

It turns out that $\left(\frac n{n^2+1}\right)_{n\in\mathbb N}$ is decreasing. Therefore, the series $\displaystyle\sum_{n=1}^\infty(-1)^n\frac n{n^2+1}$ and so the series $\displaystyle\sum_{n=0}^\infty(-1)^n\frac n{n^2+1}$ converges too.

Answer 2

As in series, a finite number of terms does not make any change in convergence. So even if for some terms, a principle is not working then you can remove that term and analyze the remaining infinitely many terms.
As in your case, you can remove the first term.
Remaining part as Santos Sir Provided.