Mathematics, 04.10.2021 18:30 raytaygirl
Lim n-> infinity [1/3 + 1/3² + 1/3³ + . . . .+ 1/3ⁿ]
Answers: 1
![\rm :\longmapsto\:\displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]](/tpl/images/2490/1626/ad95d.png)
![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{1 - \dfrac{1}{3} } \bigg]](/tpl/images/2490/1626/5f958.png)
![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{\dfrac{3 - 1}{3} } \bigg]](/tpl/images/2490/1626/dbb5e.png)
![\rm :\longmapsto\:S_n = \dfrac{1}{3} \bigg[\dfrac{1 - {\bigg[\dfrac{1}{3} \bigg]}^{n} }{\dfrac{2}{3} } \bigg]](/tpl/images/2490/1626/d56be.png)
![\bf\implies \:S_n = \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](/tpl/images/2490/1626/bfed7.png)
![\bf :\longmapsto\:\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } = \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](/tpl/images/2490/1626/3ecfb.png)
![\purple{\rm :\longmapsto\:\displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]}](/tpl/images/2490/1626/504ee.png)
![\rm \: = \: \displaystyle\lim_{n \to \infty }\rm \dfrac{1}{2}\bigg[1 - \dfrac{1}{ {3}^{n} } \bigg]](/tpl/images/2490/1626/243db.png)
![\rm \: = \: \rm \dfrac{1}{2}\bigg[1 - 0 \bigg]](/tpl/images/2490/1626/58975.png)
![\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{n \to \infty }\rm \bigg[\dfrac{1}{3} + \dfrac{1}{ {3}^{2} } + \dfrac{1}{ {3}^{3} } + - - + \dfrac{1}{ {3}^{n} } \bigg]} = \frac{1}{2}}}](/tpl/images/2490/1626/93945.png)
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Lim n-> infinity [1/3 + 1/3² + 1/3³ + . . . .+ 1/3ⁿ]...
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