The sequence is converges and the limit is [tex]\lim_{n \to \infty} S_n=51\frac{3}{7}[/tex]
Step-by-step explanation:
In the geometric sequence
- The sequence is converges if IrI < 1, where r is the constant ratio between each two consecutive terms
- The sequence is diverges if IrI > 1
IrI < 1 means → -1 < r < 1
IrI > 1 means → r < -1 and r > 1
∵ The sequence is 60 , -10 , [tex]\frac{5}{3}[/tex] , [tex]-\frac{5}{18}[/tex] , .....
∵ [tex]r=\frac{a_{2}}{a_{1}}[/tex] , where [tex]a_{1}[/tex] is the first term and
[tex]a_{2}[/tex] is the second term
∵ [tex]a_{1}[/tex] = 60
∵ [tex]a_{2}[/tex] = -10
∴ [tex]r=\frac{-10}{60}[/tex] = [tex]\frac{-1}{6}[/tex]
∴ The constant ratio is [tex]\frac{-1}{6}[/tex]
∵ -1 < [tex]\frac{-1}{6}[/tex] < 1
∴ I [tex]\frac{-1}{6}[/tex] I < 1
∴ The sequence is converges
∵ The limit is the sum to infinity
∵ [tex]\lim_{n \to \infty} S_n=\frac{a_{1}}{1-r}[/tex]
∵ a = 60 and r = [tex]\frac{-1}{6}[/tex]
- Substitute these values in the rule above
∴ [tex]\lim_{n \to \infty} S_n=\frac{60}{1-(\frac{-1}{6})}[/tex]
∴ [tex]\lim_{n \to \infty} S_n=51\frac{3}{7}[/tex]
The sequence is converges and the limit is [tex]\lim_{n \to \infty} S_n=51\frac{3}{7}[/tex]
Learn more:
You can learn more about sequences in brainly.com/question/7221312
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