Question

∑_(n=1)^∞▒〖( 1/2 )〗^2n

Answer 1

The series converges to
`(1)/(3)`

Explanation:

It seems to be this series:

`$ \sum_(n=1)^(\infty) \left((1)/(2) \right)^(2n)$`

We have

`$ \sum_(n=1)^(\infty) \left((1)/(2) \right)^(2n) = \sum_(n=1)^(\infty) \left((1)/(4) \right)^(n)$`

Using the Root test we can see that this series converges once

`$ \lim_(n \to \infty) \sqrt[n]a_n < 1 \implies \sum_(n=1)^(\infty) a_n \text{ is convergent}$`

Then,
`$\lim_(n \to \infty) \sqrt[n]{\left((1)/(4) \right)^(n)} = \lim_(n \to \infty) (1)/(4) = (1)/(4) < 1$`

The series is convergent.

Once the series is geometric, the first term is
`(1)/(4)` and the ratio is also
`(1)/(4)` in this case.

The sum of infinite geometric series is
`S = (a_1)/(1-r)` such that
`r < 1`

`\therefore S = ((1)/(4) )/(1-(1)/(4)) = (1)/(3)`