What is the sum from n=1 to infinity of 2(1/3)^{n-1} ?

The sum from n=1 to infinity of 2(1/3)^{n-1} is converges

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sum from n=1 to infinity of 2(1/3)^{n-1}

n = 1 \sum \infty 2 (\frac{1}{3})^{n - 1}

converges

Solution steps

n = 1 \sum \infty 2 (\frac{1}{3})^{n - 1}

Apply the constant multiplication rule:

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

= 2 \cdot n = 1 \sum \infty (\frac{1}{3})^{n - 1}

Apply Series Ratio Test:

converges

= 2 converges

= converges

n = 1 \sum \infty \frac{9}{n ^{2} + 25}

n = 1 \sum \infty \frac{1}{3}

n = 0 \sum \infty (- \frac{3}{7})^{n}

n = 1 \sum \infty 9 (0.8)^{n - 1}

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

What is the sum from n=1 to infinity of 2(1/3)^{n-1} ?
The sum from n=1 to infinity of 2(1/3)^{n-1} is converges