Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

Answer 1

The sequence is converges and the limit is $\lim_{n \to \infty} S_n=51\frac{3}{7}$

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 , $\frac{5}{3}$ , $-\frac{5}{18}$ , .....

∵ $r=\frac{a_{2}}{a_{1}}$ , where $a_{1}$ is the first term and

  $a_{2}$ is the second term

∵ $a_{1}$ = 60

∵ $a_{2}$ = -10

∴ $r=\frac{-10}{60}$ = $\frac{-1}{6}$

∴ The constant ratio is $\frac{-1}{6}$

∵ -1 < $\frac{-1}{6}$ < 1

∴ I $\frac{-1}{6}$ I < 1

∴ The sequence is converges

∵ The limit is the sum to infinity

∵ $\lim_{n \to \infty} S_n=\frac{a_{1}}{1-r}$

∵ a = 60 and r = $\frac{-1}{6}$

- Substitute these values in the rule above

∴ $\lim_{n \to \infty} S_n=\frac{60}{1-(\frac{-1}{6})}$

∴ $\lim_{n \to \infty} S_n=51\frac{3}{7}$

The sequence is converges and the limit is $\lim_{n \to \infty} S_n=51\frac{3}{7}$

Learn more:

You can learn more about sequences in brainly.com/question/7221312

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