What is the sum from n=0 to infinity of n/(n^2+n+1) ?

The sum from n=0 to infinity of n/(n^2+n+1) is diverges

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sum from n=0 to infinity of n/(n^2+n+1)

n = 0 \sum \infty \frac{n}{n ^{2} + n + 1}

diverges

Solution steps

n = 0 \sum \infty \frac{n}{n ^{2} + n + 1}

Apply Series Integral Test:

diverges

= diverges

n = 1 \sum \infty n (\frac{8}{9})^{n}

k = 1 \sum \infty \frac{1}{k} - \frac{1}{k + 2}

n = 1 \sum \infty 6 ln (\frac{n}{n + 1})

k = 0 \sum \infty \frac{k}{2 k + 1}

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

What is the sum from n=0 to infinity of n/(n^2+n+1) ?
The sum from n=0 to infinity of n/(n^2+n+1) is diverges