What is the sum from k=1 to infinity of (k!)/((2k)!) ?

The sum from k=1 to infinity of (k!)/((2k)!) is converges

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sum from k=1 to infinity of (k!)/((2k)!)

k = 1 \sum \infty \frac{k !}{( 2 k ) !}

converges

Solution steps

k = 1 \sum \infty \frac{k !}{( 2 k ) !}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty (\frac{- e}{π})^{n}

n = 0 \sum \infty 3 (- 1)^{n}

n = 3 \sum \infty \frac{n ^{n}}{n !}

n = 3 \sum \infty \frac{4}{n ^{2} - 4}

k = 1 \sum \infty \frac{1}{k + 3}

What is the sum from k=1 to infinity of (k!)/((2k)!) ?
The sum from k=1 to infinity of (k!)/((2k)!) is converges