# How do you determine if a_n=1-1.1+1.11-1.111+1.1111-... converge and find the sums when they exist?

Apr 27, 2017

$1.01010101 \ldots$

#### Explanation:

${a}_{n} = 1 - 1.1 + 1.11 - 1.111 + 1.1111 - \ldots$

a_n = 1 + (-1.1 + 1.11) +(- 1.111 + 1.1111) +( - ...

a_n = 1 + (0.01) +(0.0001) +( - ...

we can say that it squence is geometric progression where
${a}_{1} = 1$, common ratio, $r = \frac{0.01}{1} = \frac{0.0001}{0.01} = \frac{1}{100}$

sum of infinity, ${s}_{\infty} = {a}_{1} / \left(1 - r\right)$

${S}_{\infty} = \frac{1}{1 - \frac{1}{100}} = \frac{1}{\frac{99}{100}} = \frac{100}{99} = 1.01010101 \ldots$