What is the sum from n=0 to infinity of 2(2/e)^n ?

The sum from n=0 to infinity of 2(2/e)^n is 2*e/(e-2)

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sum from n=0 to infinity of 2(2/e)^n

n = 0 \sum \infty 2 (\frac{2}{e})^{n}

2 \cdot \frac{e}{e - 2}

Decimal

7.56884 \dots

Solution steps

n = 0 \sum \infty 2 (\frac{2}{e})^{n}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= 2 \cdot n = 0 \sum \infty (\frac{2}{e})^{n}

Apply Series Geometric Test:

\frac{e}{e - 2}

= 2 \cdot \frac{e}{e - 2}

What is the sum from n=0 to infinity of 2(2/e)^n ?
The sum from n=0 to infinity of 2(2/e)^n is 2*e/(e-2)