Question

I have a question about geometric sequences with infinite sums picture included

Answer 1

Answer:
`\begin{gathered} a)\text{ S}_n\text{ = -25} \\ The\text{ series converges} \end{gathered}`Step-by-step explanation:
`\begin{gathered} Given:\text{ } \\ \text{a = -5} \\ r\text{ = 4/5} \end{gathered}`

The formula relating the sum of infinite series of a geometric sequence:

`\sum_{n\mathop{=}1}^(\infty)a_n\text{ = }(a)/(1-r)`
`\begin{gathered} The\text{ sum of infinite series of the Geometric sequence = }(a)/(1-r) \\ substitute\text{ the values:} \\ S_n\text{ = }(-5)/(1-(4)/(5)) \\ S_n\text{ = }(-5)/((5-4)/(5)) \\ S_n\text{ = }(-5)/((1)/(5)) \\ S_n\text{ = -5 }*\text{ }(5)/(1)\text{ = -25} \\ \\ S_n\text{ = S}_(\infty)\text{ = -25} \\ \\ S_(\infty_)\text{ = -25} \end{gathered}`

For the series to converge:

`\begin{gathered} For\text{ the series to converge:} \\ \lim_(n\to\infty)S_n\text{ =}\sum_{n\mathop{=}1}^(\infty)a_n\text{ = actual value} \\ \\ For\text{ the series to diverge:} \\ \lim_(n\to\infty)S_n\text{ = }\pm\infty \\ \\ Also\text{ if \mid r\mid < 1, it converges} \\ if\text{ \mid r\mid \ge 1, it diverges} \end{gathered}`