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Question Number 120810 by fajri last updated on 03/Nov/20

$$Determinetheconvergenceintervvalof:$$$$\underset{n=0}{\sum ^{\mathrm{\infty }}}{(-1)}^n{(x-1)}^n$$

Answered by 675480065 last updated on 03/Nov/20

$$\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\mid \frac{{\mathrm{U}}_{\mathrm{n}+1}}{{\mathrm{U}}_{\mathrm{n}}}\mid <1$$$$\Rightarrow \underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\mid \frac{{(-1)}^{\mathrm{n}+1}{(\mathrm{x}-1)}^{\mathrm{n}+1}}{{(-1)}^{\mathrm{n}}{(\mathrm{x}-1)}^{\mathrm{n}}}\mid <1$$$$\Rightarrow \underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\mid \frac{-{(-1)}^{\mathrm{n}}{(\mathrm{x}-1)}^{\mathrm{n}}(\mathrm{x}-1)}{{(-1)}^{\mathrm{n}}{(\mathrm{x}-1)}^{\mathrm{n}}}\mid <1$$$$\Rightarrow \mid \mathrm{x}-1\mid <2\Rightarrow -2<\mathrm{x}-1<2$$$$\mathrm{hence}-1<\mathrm{x}<3$$

Commented by fajri last updated on 03/Nov/20

$$thankssir...$$

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