What is the sum from n=1 to infinity of (n!)/((4n)!) ?

The sum from n=1 to infinity of (n!)/((4n)!) is converges

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sum from n=1 to infinity of (n!)/((4n)!)

n = 1 \sum \infty \frac{n !}{( 4 n ) !}

converges

Solution steps

n = 1 \sum \infty \frac{n !}{( 4 n ) !}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty (\frac{7}{5})^{n}

n = 1 \sum \infty \frac{1}{12 n + 13}

n = 1 \sum \infty 5 n + 2

n = 0 \sum \infty \frac{3}{2} (\frac{1}{2})^{n}

n = 1 \sum \infty \frac{3}{( 4 n + 2 ) ^{3}}

What is the sum from n=1 to infinity of (n!)/((4n)!) ?
The sum from n=1 to infinity of (n!)/((4n)!) is converges