Question

How do you find the sum of the infinite geometric series -5 – 5/2 – 5/4 - . . .?

Answer 1

`(-5)+(-5/2)+(-5/4)+(-5/8)+...=color(blue)(-10)`

Explanation:

Consider the geometric series defined as
`color(white)("XXX")a_n=1/(2^n)`

`Sigma_(i=0)^oo a_i=2`
`color(white)("XXXXXXXXXXX")`[see below, if necessary for why this is true]

Each term of the given geometric series is simply `(-5)` time the corresponding term of `a_n`.
Therefore `Sigma ((-5)+(-5/2)+(-5/4)+...)`

`color(white)("XXXXXX")=Sigma_(i=0)^oo (-5) * (a_i)`

`color(white)("XXXXXX")=(-5) * Sigma_(i=0)^oo a_i`

`color(white)("XXXXXX") = (-5) * 2`

`color(white)("XXXXXX") = -10`

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Why is `Sigma_(i=0)^oo 1/(2^i)=2`?

Notice that `a_0=1`
and each additional term `a_i` reduces the distance between the sum up to this point and `2` by `1/2`