Use the Integral Test to determine the convergence or divergence of the series. $$\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }$$
A) diverges
B) Integral Test inconclusive
C) converges

Question

Quizplus · Accepted Answer

We need to use the Integral Test, which states that if $f(x)$ is a positive, continuous, and decreasing function on the interval $[n,\infty)$, then the series $\sum_{n=1}^\infty f(n)$ converges if and only if the improper integral $\int_n^\infty f(x)\,dx$ converges.

In this case, we have $f(n) = \frac{\ln n}{n^{10}}$, which is positive and decreasing for $n\geq 2$. So, we can apply the Integral Test:

$$\int_2^\infty \frac{\ln x}{x^{10}}\,dx = \left[-\frac{\ln x}{9x^9}ight]_2^\infty + \int_2^\infty \frac{1}{9x^{10}}\,dx$$

The first term evaluates to zero (the limit as $x	o\infty$ is zero), and the second term evaluates to $\left[-\frac{1}{81x^9}ight]_2^\infty = \frac{1}{81\cdot 2^9}$. Since this integral converges, so does the series $\sum_{n=2}^\infty \frac{\ln n}{n^{10}}$.

Therefore, the answer is choice C, converges.

Use the Integral Test to Determine the Convergence or Divergence ∑n=2∞ln⁡nn10\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }n=2∑∞​n10lnn​

Use the Integral Test to Determine the Convergence or Divergence $\sum _ { n = 2 } ^ { \infty } \frac { \ln n } { n ^ { 10 } }$