# Question regarding infinite series and their convergences or divergences.

If I have an infinite series $a_n$ such as $\frac{1 + 4^n}{1 + 3^n}$ and $a_n$ is equal to the infinite series $b_n + c_n$ or $\frac{1}{3^n} + \frac{4^n}{3^n}$, is it possible to show in a question that the series $a(n)$ diverges by showing that the series $c(n)$ or $\frac{4^n}{3^n}$ diverges? I was doing a question and this is how I showed it. But when I checked the answer in my textbook they proved it by comparing the series $a_n$ to the different series $\frac{1 + 4^n}{3^n}$ and found the limit of the ratio of those series. That method also makes sense to me too but I find my way easier but I don't know if my way is allowed? My way makes sense to me because if I have a series and it's equal to $a + b$ of course both a and b have to converge for the whole series to converse. You can't have part of the series converge and the other part diverge. Could someone clear my confusion. Thank You.

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One common sense rule (which you intuited): $\sum a_n$ diverges $\implies \sum a_n+b_n$ diverges provided that $a_n,b_n$ are of the same parity $\forall n$. This is far from necessary or sufficient conditions, but it accords with your intuitive idea. Note however that if the $a_n,b_n$ are of different parity, then even though one of the sequences may diverge on its own, the sum may "cancel" in some way and divergence will not be guaranteed. – Coffee_Table Mar 24 '13 at 3:32