Divergence/Convergency of a series

[Heaviplace]

Homework Statement

I have to test this series for convergency/divergency

Ʃ1000(3/4)^n - n/(n+10000)

n=1 to ∞

The Attempt at a Solution

At first I thought of analyzing both parts separately, (3/4)^n is a geometric series with r < 1 and so it converges, and applying the root test to n/(n+10000). Then I'd have the sum of two convergent series that ofc, also converges.

Went to test my hypothesis on WolframAlpha, and it actually told me the series diverges by the limit test, which I verified to be true.

So my question is, why was the method I used, wrong?

Thanks in advance!

 

[Dick]

n/(n+10000) doesn't converge by the root test. It doesn't converge at all. It doesn't even approach zero!

 

[Heaviplace]

Indeed, you are right!

But that still leaves me the problem, but this time with the sum of a convergent series and divergent series =/

 

[Dick]

Indeed, you are right!

But that still leaves me the problem, but this time with the sum of a convergent series and divergent series =/

The sum of a convergent series and a divergent series always diverges. Think of the partial sums.

 

[Heaviplace]

Ok, see if I got it right, since one of the series does not converge, I can't bring up the property of the sum of two convergent series.

And even though (3/4)^n is convergent, the limit test shows that the sum of both series (<>0) is not.

 

[Heaviplace]

Thanks really man, you really cleared things up for me.

Just a last question a bit off topic, where's the edit button? =P

 

[Dick]

Ok, see if I got it right, since one of the series does not converge, I can't bring up the property of the sum of two convergent series.

And even though (3/4)^n is convergent, the limit test shows that the sum of both series (<>0) is not.

Sure, you can think of it that way. (3/4)^n approaches zero as n->infinity. n/(n+10000) doesn't. So the sum doesn't approach zero. Doesn't converge by the limit test.

 

[Dick]

Thanks really man, you really cleared things up for me.

Just a last question a bit off topic, where's the edit button? =P

The 'Edit' button has been coming and going lately. No idea why.