Answer to Question #303616 in Real Analysis for Dhruv rawat

Question #303616

Find whether the following series are convergent or not


i. ∞Σ n=1 (3n-1)/7^n

1
Expert's answer
2022-02-28T15:34:22-0500

Use the Ratio Test


an+1an=3(n+1)17n+13n17n|\dfrac{a_{n+1}}{a_n}|=\dfrac{\dfrac{3(n+1)-1}{7^{n+1}}}{\dfrac{3n-1}{7^n}}

=173n+23n117 as n=\dfrac{1}{7}\cdot \dfrac{3n+2}{3n-1}\to\dfrac{1}{7}\ as\ n\to \infin



Since 17<1,\dfrac{1}{7}<1, then the series n=13n17n\displaystyle\sum_{n=1}^{\infin}\dfrac{3n-1}{7^n} is convergent by the Ratio Test.




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