Answered: 1 1 41. dx (x+2)(In(x+2))3 2(In(3))2 :…

Math

Calculus

1 1 41. dx (x+2)(In(x+2))3 2(In(3))2 : so the series converges.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 44E

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Question

1
1
41.
dx
(x+2)(In(x+2))3
2(In(3))2 :
so the series converges.

Expert Solution

Step 1

Integral test:

Let f(x) is continuous, positive and decreasing function on $[k, \infty)$ and $f (n) = a_{n}$ then

If $\int_{k}^{\infty} f (x) d x$ is convergent so $\sum_{n = k}^{\infty} a_{n}$ is also convergent

$[\int_{k}^{\infty} f (x) d x i s c o n v e r g e n t w h e n \int_{k}^{\infty} f (x) d x i s a n y f i n i t e n u m b e r]$

If $\int_{k}^{\infty} f (x) d x$ is divergent so $\sum_{n = k}^{\infty} a_{n}$ is also divergent

$[\int_{k}^{\infty} f (x) d x i s d i v e r g e n t w h e n \int_{k}^{\infty} f (x) d x = \infty]$

Step 2

Given that

$\int_{1}^{\infty} \frac{1}{(x + 2) {(\ln (x + 2))}^{3}} d x = \frac{1}{2 {(\ln (3))}^{2}}$

Here k=1 and $f (x) = \frac{1}{(x + 2) {(\ln (x + 2))}^{3}}$

And $\int_{1}^{\infty} \frac{1}{(x + 2) {(\ln (x + 2))}^{3}} d x = \frac{1}{2 {(\ln (3))}^{2}} = f i n i t e v a l u e$

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