How do I prove this series is decreasing?

Question

I need to prove this series
(-1)^(n+1) * (n^2) / [(n^3)+4]
is convergent or divergent.
I did the intergral test for (n^2) / [(n^3)+4] and it diverged.
So then I took the limit as n goes to infinity for (n^2) / [(n^3)+4] and got zero.
I know this series decreases but I can't figure out how to prove it.
Please help.
Should I do:
An+1 - An < 0
(An+1)/An < 1
An' < 0

Answer 1

The idea of taking the derivative the nth term and showing it's neg looks good to me, but if you want an algebraic proof:
Let's start with
1/[n +4/n^2)] and compare it to 1/[(n + 1) + 4/(n + 1)^2]
We want to show the second is smaller than the first.
We want to show denom of the second is greater than the first
n + 1 + 4/(n + 1)^2 as compared to n + 4/n^2?
or 1 + 4/(n+ 1 )^2 against 4/n^2?
or 1 against 4/ n - 4/ (n + 1)^2 ?
For n > 4, 4/n < 1 so the right hand side of the line just
above this one gives us 1 > 4/ n - 4/ (n + 1)^2
Working your way backward you get the denom of the second is greater than the first which means the n + 1 st
term is less that the n th.

Answer 2

Since the sequence alternates between positive and negative numbers, the series will alternately go up and down. It is not decreasing.

Note that each for each term, An, you have:

-1/n < An < 1/n

You also have that An > 1/2n for positive terms and An < -1/2n for negative terms. You should be able to use these facts to show your point by looking at whether or not the sum of (-1)^(n+1)*(1/n) converges.

Answer 3

you can prove a function is decreasing by looking at the derivative. If it is negative then it is decreasing.

The derived function is (-n^4 + 8n) / (n^3 + 4) ^2

This function is negative for n> 2 hence the series starts decreasing for n>=2, which is all that is needed ( ultimately decreasing)

EDIT: Taranto: she is using liebniz theorem to prove conditional convergence and is looking at the series of absolute values!!!!!

Answer 4

attempt an alternating sequence the position a_n = 2k+a million / (n85f81286df9f355c5c917c0dac92985f81286df9f355c5c917c0dac929) if n = even and 2k+a million/ n if n = spectacular, write the 85f81286df9f355c5c917c0dac929st 8 words, including each and every 2 jointly, if executed wisely you need to get ? 2k+a million / [85f81286df9f355c5c917c0dac92985f81286df9f355c5c917c0dac92985f81286df9f355c5c917c0dac92985f81286df9f355c5c917c0dac929] , 2k+a million = 0,2k+a million,2k+a million,.....which diverges even although a_n ---> 0