Answer
Convergent
Work Step by Step
Alternating series Test: Let us suppose a series $a_n$ such that $a_n=(-1)^{n+1} b_n$ or $a_n=(-1)^n b_n$ when $b_n \geq 0$ for all $n$.
1) $\lim\limits_{n \to \infty}b_n=0$
2) $b_n$ shows a decreasing sequence.
For a series to be convergent, the above two conditions must be satisfied.
We have: $b_k=\dfrac{1}{2k+1}$
1) We see that $\lim\limits_{k \to \infty}b_k=\lim\limits_{k \to \infty} \dfrac{1}{2k+1}\\=\dfrac{1}{\infty}\\=0$
(2) In $b_k$, the denominator is increasing, so $b_k$ shows a decreasing sequence.
We conclude that both conditions of the Alternating series Test are satisfied, so the given series is convergent.
