What is the sum from n=0 to infinity of ((n+5)/n)^n ?

The sum from n=0 to infinity of ((n+5)/n)^n is diverges

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sum from n=0 to infinity of ((n+5)/n)^n

n = 0 \sum \infty (\frac{n + 5}{n})^{n}

diverges

Solution steps

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

(\frac{n + 5}{n}) = (\frac{n}{n} + \frac{5}{n})

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

Simplify

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

Apply Series Divergence Test:

diverges

= diverges

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

n = 1 \sum \infty \frac{1}{1 + 8 n}

n = 0 \sum \infty \frac{3}{n}

n = 1 \sum \infty 4 - \frac{1}{n}

n = 0 \sum \infty \frac{7}{1 0 ^{- n}}

What is the sum from n=0 to infinity of ((n+5)/n)^n ?
The sum from n=0 to infinity of ((n+5)/n)^n is diverges