Does the Series converge? If so find the sum

The series given in the problem is `sum_(n=1)^oo e^(-2n)` .

This is a convergent series.

`sum_(n=1)^oo e^(-2n)`

= `sum_(n=1)^oo 1/e^(2n)`

This is a geometric series with first term `a = 1/e^2` and common ratio `r = 1/e^2` .

The sum `sum_(n=1)^oo e^(-2n) = (1/e^2)/(1 - 1/e^2)`

= `(1/e^2)/(e^2/e^2 - 1/e^2)`

= `(1/e^2)/((e^2 - 1)/e^2)`

= `1/(e^2 - 1)`

The required sum of the series `sum_(n=1)^oo e^(-2n) = 1/(e^2 - 1)`

Comments

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