# What is the sum of the infinite geometric series #sum_(n=0)^oo(1/e)^n# ?

We are given the infinite geometric series :

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#sum_(n=0)^oo (1/e)^n = e/(e-1)#

The general term of a geometric series can be written:

so the sum is:

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The sum of the infinite geometric series ( \sum_{n=0}^{\infty} \left(\frac{1}{e}\right)^n ) can be found using the formula for the sum of an infinite geometric series, which is given by:

[ S = \frac{a}{1 - r} ]

Where ( a ) is the first term of the series and ( r ) is the common ratio. In this case, the first term ( a ) is ( 1 ) (since ( n = 0 )) and the common ratio ( r ) is ( \frac{1}{e} ).

Substitute these values into the formula:

[ S = \frac{1}{1 - \frac{1}{e}} ]

Simplify the expression:

[ S = \frac{1}{1 - \frac{1}{e}} = \frac{1}{\frac{e - 1}{e}} = \frac{e}{e - 1} ]

So, the sum of the infinite geometric series ( \sum_{n=0}^{\infty} \left(\frac{1}{e}\right)^n ) is ( \frac{e}{e - 1} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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