Question

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

Answer 1

`1.01010101...`

Explanation:

`a_n = 1 -1.1 + 1.11 - 1.111 + 1.1111 - ...`

`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 = 0.01/1 =0.0001/0.01 = 1/100`

sum of infinity, `s_oo = a_1/(1-r)`

`S_oo= 1/(1 - 1/100) = 1/(99/100) = 100/99 = 1.01010101...`