## Thomas' Calculus 13th Edition

Let us consider $a_n=\dfrac{n^{10}}{10^n}$ In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges. Then $L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^{10}}{10^{n+1}}}{\dfrac{n^{10}}{10^n}}|$ $\implies L=\dfrac{1}{10} \cdot \lim\limits_{n \to \infty}(\dfrac{n+1}{n})^{10}=\dfrac{1}{10} \cdot 1=\dfrac{1}{10} \lt 1$ Hence, the series Converges by the ratio test.