What is the sum from n=0 to infinity of (2n)/(n^3+1) ?

The sum from n=0 to infinity of (2n)/(n^3+1) is converges

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sum from n=0 to infinity of (2n)/(n^3+1)

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

converges

Solution steps

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

Apply the constant multiplication rule:

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

= 2 \cdot n = 0 \sum \infty \frac{n}{n ^{3} + 1}

Apply Series Comparison Test:

converges

= 2 converges

= converges

n = 1 \sum \infty \frac{1}{24} (\frac{3}{4})^{n}

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

k = 0 \sum \infty \frac{1}{k ^{2} + 4}

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

n = 0 \sum \infty (\frac{n}{4 - n})^{3}

What is the sum from n=0 to infinity of (2n)/(n^3+1) ?
The sum from n=0 to infinity of (2n)/(n^3+1) is converges