Question

Comparison test to determine if improper integral converges or diverges, (3+cos(x))/(x^3) from 1 to infinity

Answer 1

3+cos(x) varies between 4 and 2 as x approaches infinity. Since 4 >= 3+cos(x) always, the series of (3+cos(x))/x^3 will converge if 4/x^3 converges.

Answer 2

f(x) >= g(x) >= 0
f(x) = 1/x^2
g(x) = (3+cos(x))/(x^3)
because of f(x) is larger for all values of g(x) and f(x) converges to \[\int\limits_{1}^{∞} 1/x^2 dx = \lim t \rightarrow∞ \int\limits_{1}^{t} 1/x^2 = {-1/x}\]
from t to1 = 1

Answer 3

1/x^2 is not larger than (3+cos(x))(x^3) For quite a few values greater than 1.

Answer 4

Infinitely many in fact.

Answer 5

I dont understand how he got the 1/x^2 :P , also whats the difference if you select 2 instead of 4? say 2<=3+cos(x)?

Answer 6

You want to select something that is greater than the function you want to solve. If you can find something which converges but which is strictly larger than the function you're looking to solve, then you know that your function must also converge.

Answer 7

Oh ok, so when I get 4/x^3 do I have to evaluate the definite integral of that and find out if it converges or diverges?

Answer 8

Oops i meant (3+cos(x))/ x^2 which is greater then all values of (3+cos(x))/ x^2

Answer 9

***(3+cos(x))/ x^3 sorry

Answer 10

Ok nevermind, i got it, thanks!

Answer 11

Yep. the point is to find a function that is infinitely larger and if that function converges then the smaller one converges. However if the function ( f(x) for example) you test is divergent then f(x) >= g(x) >= 0 then g(x) can't be said is divergent. but if the function ( g(x) in this case) you test for is smaller and divergent then f(x) is divergent.

Answer 12

so what if the interval was not to infinity?

Answer 13

if the function is convergent on the interval and larger (meaning on discontinuities), the function is only to be said convergent on that interval.

Answer 14

No*