Answer to Question #303616 in Real Analysis for Dhruv rawat

Question

Find whether the following series are convergent or not

i. ∞Σ n=1 (3n-1)/7^n

Answer 1

Use the Ratio Test

$$|\dfrac{a_{n+1}}{a_n}|=\dfrac{\dfrac{3(n+1)-1}{7^{n+1}}}{\dfrac{3n-1}{7^n}}$$

$$=\dfrac{1}{7}\cdot \dfrac{3n+2}{3n-1}\to\dfrac{1}{7}\ as\ n\to \infin$$

Since $\dfrac{1}{7}<1,$ then the series $\displaystyle\sum_{n=1}^{\infin}\dfrac{3n-1}{7^n}$ is convergent by the Ratio Test.