Question

show that the series diverges

Answer 1

\[\sum_{n=1}^{\infty} \frac{ n^2}{ 5n^2+4 }\]

Answer 2

hello could you help me?

Answer 3

do you know the "ratio test" ?

Answer 4

no but could we do it doing the geometric test?

Answer 5

ratio test is easy
http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx

Answer 6

i see

Answer 7

so i would need to cross multiply right

Answer 8

I'm testing to see if it's applicable here.

Answer 9

Here is a list of tests.
https://en.wikipedia.org/wiki/Convergence_tests#List_of_tests
the very first one works
the limit of the expression
\[ \lim_{n \rightarrow \infty}\frac{ n^2}{ 5n^2+4 } = ?\]

Answer 10

to find the limit, divide top and bottom by n^2 to get
\[ \frac{1}{5 + \frac{4}{n^2}} \]
as n goes to infinity 4/n^2 goes to 0, and the expression to 1/5

Answer 11

the limit of the summand one?

Answer 12

and the an infinite sum of 1/5's diverges to infinity

yes, the "limit of summand" test

Answer 13

but wouldnt it be 1/9 because the 4/(1)^2 is 4 and 1/5+4 is 1/9?

Answer 14

dont you replace n with the 1

Answer 15

the problem is to show that as we add up the series (of numbers) the sum gets larger and larger (as large as you want if n is big enough), i.e. show the sum diverges.

to do that, there are tests.
the limit test says: find what number n^2/(5n^2 +4) approaches as n gets large.

Answer 16

in other words, when n is a gazillion , what is the approximate value of
n^2 / (5n^2 + 4) ?

Answer 17

2

Answer 18

how did you get 2?

Answer 19

I think the divergence test might be a pretty nice choice as well to consider.

Answer 20

no its saying .2

Answer 21

yes 0.2 is the limit
that means if you go out "far enough" you can imagine you are adding 1/5 + 1/5 + 1/5 +...
forever.

it should be clear that sum will go to infinity, right ? i.e. diverges.

Answer 22

yeah it much clearer now thanks phil