Answered: "Use the Integral Test to determine whether the..."

gonjenjemeb

Answered question

2021-11-12

Use the Integral Test to determine whether the series is convergent or divergent.

\sum_{n = 1}^{\infty} \frac{n}{n^{2} + 1}

Florence Pittman

Beginner2021-11-13Added 15 answers

\sum_{n = 1}^{\infty} \frac{n}{n^{2} + 1}

f (x) = \frac{x}{x^{2} + 1}

is decrasing since the degree of the denominator is higher than the degree of the numerator. It is also positive and continuous for

x \geq 0

so we can use the Integral Test.

\int_{1}^{\infty} \frac{x}{x^{2} + 1} d x = lim_{b \to \infty} \frac{x}{x^{2} + 1} d x

= lim_{b \to \infty} \frac{1}{2} \ln (x^{2} + 1) ∣_{1}^{b}

= lim_{b \to \infty} \frac{1}{2} (\ln (b^{2} + 1) - \ln (2))

= \infty

The integral diverges, so by the Integral Test, the series also diverges.
Result: divergent

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