Is Convergence of Series Possible w/ Divergent Parts?

[manenbu]

I have an infinite series, let's say Σa_n, and I can split it into Σb_n + Σc_n. If one of the series is convergent (let's say Σb_n) while the other is divergent (Σc_n), is it safe to say that the original series (Σa_n) is also divergent?

If to show an example, I have:
$$\sum {\frac{1+n}{1+n^2}} = \sum {\frac{1}{1+n^2}} + \sum {\frac{n}{1+n^2}}$$
Intuitively, I can say that yes, but is this enough?
Or maybe just for cases where both of the series are either negative or positive (positive in this case)?

 

[Office_Shredder]

I have an infinite series, let's say Σa_n, and I can split it into Σb_n + Σc_n. If one of the series is convergent (let's say Σb_n) while the other is divergent (Σc_n), is it safe to say that the original series (Σa_n) is also divergent?

If to show an example, I have:
$$\sum {\frac{1+n}{1+n^2}} = \sum {\frac{1}{1+n^2}} + \sum {\frac{n}{1+n^2}}$$
Intuitively, I can say that yes, but is this enough?
Or maybe just for cases where both of the series are either negative or positive (positive in this case)?

If

Σa_n = Σb_n + Σc_n

where the b series is convergent and the c series is divergent, you can think of it like this:

Σa_n - Σb_n = Σc_n

And if the series Σa_n is convergent, then you can combine that with the series Σb_n) and get that Σc_n is convergent. But it's not, so Σa_n must be divergent