Finding the divergence or convergence of a series

[aimsanko]

Ʃ ,n=1,∞, (2/n^2+n)

Does this series converge or diverge?

Im not sure how to start can i use the comparison test here?

 

[LCKurtz]

You can begin by making sure you stated the problem you meant to state. Is that expression (2/n²)+n, which is what you wrote would mean, or 2/( n²+n)?

What have you tried? What are your thoughts about it?

 

[aimsanko]

its 2/((n^2)+n)
and I'm thinking it converges because you are adding an infinite amount of numbers that are continually getting smaller and smaller

 

[LCKurtz]

its 2/((n^2)+n)
and I'm thinking it converges because you are adding an infinite amount of numbers that are continually getting smaller and smaller

Both convergent and divergent series have an infinite number of terms that get smaller and smaller, so that observation is of no value. For example, do you know the p series
\sum \frac 1 {n^p}
and for what values of p it converges or diverges?

Do you know how to use the comparison tests? What might you compare your series with and why?

 

[aimsanko]

if p is greater than 1 the series converges
im not very good at the comparison test and how to find what to cop are it to

 

[LCKurtz]

if p is greater than 1 the series converges
im not very good at the comparison test and how to find what to cop are it to

The first step in a comparison test is to decide by examining your series whether you think it probably converges or diverges. This is usually decided by previous experience with previously analyzed series such as a p series. The next step depends on your expectation of convergence or divergence.

If you think your series might converge, try to find a larger series that you know does converge. Then you have a series smaller than a known convergent series so it converges.

If you think your series might diverge, see if you can find a smaller series that you know diverges. If your series if larger than a known divergent series, it must diverge.

Your problem has similarities to a p series about which you know convergence or divergence. So think about that.