Answer
Converges
Work Step by Step
Let us consider an alternating series $\Sigma_{k=1}^\infty(-1)^k a_k$ that can be convergent , when it satisfy the following two conditions such as: 1) The magnitude of terms must form a non-increasing sequence.
2) $\lim\limits_{k \to \infty} a_k=0$
Here, in the problem we have $a_k=\dfrac{1}{k}+\dfrac{2k^4}{k^{10}+1}$
a) In the given sequence, $a_k=\dfrac{1}{k}+\dfrac{2k^4}{k^{10}+1}$ , and $a_{k+1}=\dfrac{1}{k+1}+\dfrac{2(k+1)^4}{(k+1)^{10}+1}$ and $\dfrac{1}{k+1}+\dfrac{2(k+1)^4}{(k+1)^{10}+1} \leq \dfrac{1}{k}+\dfrac{2k^4}{k^{10}+1}$
This implies that $a_{k+1} \lt a_k$ and $a_{k+1} \gt 0$ , so the terms are a non-increasing sequence.
b) $\lim\limits_{k \to \infty} a_k=\lim\limits_{k \to \infty} (\dfrac{1}{k}+\dfrac{2k^4}{k^{10}+1})= \lim\limits_{k \to \infty} \dfrac{1}{k}+\lim\limits_{k \to \infty} \dfrac{2k^4}{k^{10}+1}=0$
This implies that the given series converges.
