Answered

Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

MY NOTES
lim
n18
ASK YOUR TEACHER
Use the Limit Comparison Test to determine the convergence or divergence of the series.
00
n=1
n3
converges
O diverges
n+ 3
n+ 3
n3-5n+8
- 5n+ 8
Need Help?
-
=L>0
Read It
9,433
PRACTICE ANOTHER
Watch It
AUG
3
tvā™«ā™¬
(a
A

MY NOTES Lim N18 ASK YOUR TEACHER Use The Limit Comparison Test To Determine The Convergence Or Divergence Of The Series 00 N1 N3 Converges O Diverges N 3 N 3 N class=

Sagot :

Answers:

Part 1) Ā [tex]\displaystyle \frac{1}{n^2}[/tex] or 1/(n^2) goes in the box.

Part 2) Ā The series converges

======================================================

Work Shown:

[tex]\displaystyle L = \lim_{n\to \infty} \frac{a_n}{b_n}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{\frac{n+3}{n^3-5n+8}}{\frac{1}{n^2}}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{n+3}{n^3-5n+8}\times\frac{n^2}{1}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{n^3+3n^2}{n^3-5n+8}\\\\\\[/tex]

[tex]\displaystyle L = \lim_{n\to \infty} \frac{\frac{n^3}{n^3}+\frac{3n^2}{n^3}}{\frac{n^3}{n^3}-\frac{5n}{n^3}+\frac{8}{n^3}}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{1+\frac{3}{n}}{1-\frac{5}{n^2}+\frac{8}{n^3}}\\\\\\\displaystyle L = \frac{1+0}{1-0+0}\\\\\\\displaystyle L = 1\\\\\\[/tex]

Because L is finite and positive, i.e. [tex]0 < L < \infty[/tex], this means that the original series given and the series Ā 

[tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}[/tex] Ā 

either converge together or diverge together due to the Limit Comparison Test.

But the simpler series is known to converge (p-series test).

Therefore, the original series converges as well.

The two series likely don't converge to the same value, but they both converge nonetheless.

Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.