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College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.CT: Chapter Test

Problem 15CT

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Question

Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

Expert Solution

Step 1

we have the given series is,

$= \sum_{n = 2}^{\infty} \frac{\ln (n)}{n}$

Step 2

before getting into the integral test, we must assure two things first : for the integral step to apply to

$\sum_{n = N}^{\infty} f (n)$ we must have _{^{$a_{n} > 0$}}for the given interval and f(n) must be decresing on the same interval.

both of these are true since $\ln (n) > 0 a n d n > 0 o n n \in [2, \infty), s o \frac{\ln (n)}{n} > 0 o n t h e s a m e i n t e r v a l$ .

Furthermore,ln(n) grows slower than the n so we see that $\frac{\ln (n)}{n}$ is decresing because n overpowers ln(n) in the numerator.

we can also show this by taking derivative of $\frac{\ln (n)}{n}$ and showing it's away negative on $n \in [2, \infty)$ .

so wee see the integral test applies.

The integral test states that if the two if two condition are met,then for $\sum_{n = N}^{\infty} f (n)$ , evaluate the improper integral $\int_{N}^{\infty} f (x) d x$ .

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