Find the sum of the following infinite geometric series, if it exists. 1/2 + (-1/4) + 1/8 + (-1/16)+ ...

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to notice that the common ratio of the given geomteric sequence is `-1/2` and you may evaluate such that:

`(-1/4)/(1/2) = (1/8)/(-1/4) = -(1/16)/(1/8) = ... = -1/2 = r`

You should use the following formula for summation of `n` terms of geometric sequence, such that:

`S_n = (1/2)*(1 - (-1/2)^n)/(1 - (-1/2))`

`S_n = (1/2)*(1 - (-1/2)^n)/(3/2)`

`S_n = (1/3)*(1 - (-1/2)^n)`

Since the number of terms is infinite, hence `n -> oo` yields:

`lim_(n->oo) S_n = lim_(n->oo) (1/3)*(1 - (-1/2)^n)`

`lim_(n->oo) S_n = (1/3)*(1 - lim_(n->oo) (-1/2)^n)`

`lim_(n->oo) S_n = (1/3)*(1 +- 0)`

`lim_(n->oo) S_n = (1/3)`

Hence, evaluating requested sum of the given infinite geometric seqeunce, yields `lim_(n->oo) S_n = (1/3).`

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