7. Determine if the series converges. If it converges, find the sum:3n-1+2En=1 5"

∑n=1∞3n-1+25n=∑n=1∞3n-15n+∑n=1∞25n=13∑n=1∞3n5n+2∑n=1∞15n=13∑n=1∞35n+2∑n=1∞15n∑n=1∞35n is a geometric…

Answered: 7. Determine if the series converges.…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.4: Series And Their Notations

Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+

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7. Determine if the series converges. If it converges, find the sum:
3n-1+2
En=1 5"

Expert Solution

Step 1

$\sum_{n = 1}^{\infty} \frac{3^{n - 1} + 2}{5^{n}} = \sum_{n = 1}^{\infty} \frac{3^{n - 1}}{5^{n}} + \sum_{n = 1}^{\infty} \frac{2}{5^{n}} = \frac{1}{3} \sum_{n = 1}^{\infty} \frac{3^{n}}{5^{n}} + 2 \sum_{n = 1}^{\infty} \frac{1}{5^{n}} = \frac{1}{3} \sum_{n = 1}^{\infty} {(\frac{3}{5})}^{n} + 2 \sum_{n = 1}^{\infty} {(\frac{1}{5})}^{n} \sum_{n = 1}^{\infty} {(\frac{3}{5})}^{n} is a geometric series with r = \frac{3}{5} < 1 . So this series converges . \sum_{n = 1}^{\infty} {(\frac{1}{5})}^{n} is a geometric series with r = \frac{1}{5} < 1 . So this series converges . Since both part converges, that means the entire sum \sum_{n = 1}^{\infty} \frac{3^{n - 1} + 2}{5^{n}} converges .$

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