Answer
converges
Work Step by Step
For alternating series $\Sigma_{k=1}^{\infty} (-1)^{k} a_k$ (let us consider), to be convergent , we must follow the two conditions such as: a) The magnitude of terms must form a non-increasing sequence.
b) $\lim\limits_{k \to \infty}a_k=0$
We are given that $a_k=\dfrac{1}{2k+1}$
1) In the given sequence, $a_k=\dfrac{1}{2k+1}$ , the denominator increases and the numerator remains constant, so it is a non-increasing sequence.
2) $\lim\limits_{k \to \infty} \dfrac{1}{2k+1}=0$
This means that the given sequence converges.
