What is the sum from n=1 to infinity of 2/(e^n) ?

The sum from n=1 to infinity of 2/(e^n) is 2(e/(e-1)-1)

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sum from n=1 to infinity of 2/(e^n)

n = 1 \sum \infty \frac{2}{e ^{n}}

2 (\frac{e}{e - 1} - 1)

Solution steps

n = 1 \sum \infty \frac{2}{e ^{n}}

Apply the constant multiplication rule:

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

= 2 \cdot n = 1 \sum \infty \frac{1}{e ^{n}}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

= 2 (n = 0 \sum \infty \frac{1}{e ^{n}} - \frac{1}{e ^{0}})

n = 0 \sum \infty \frac{1}{e ^{n}} = \frac{e}{e - 1}

= 2 (\frac{e}{e - 1} - \frac{1}{e ^{0}})

Simplify

= 2 (\frac{e}{e - 1} - 1)

n = 1 \sum \infty (- 1)^{n} + 1

n = 1 \sum \infty (\frac{5}{3})^{n}

n = 1 \sum \infty (\frac{7 π}{8})^{n}

n = 1 \sum \infty \frac{2 n - 1}{2 ^{n}}

n = 1 \sum \infty \frac{1}{ln ( n ^{n} )}

What is the sum from n=1 to infinity of 2/(e^n) ?
The sum from n=1 to infinity of 2/(e^n) is 2(e/(e-1)-1)