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$$If \sum a_n \text{ and } \sum b_n \text{ are two positive convergent series whose sums are } a \text{ and }b, \text{ show that the series } \sum a_n+b_n \text{ is convergent.} $$

Found this question while looking at exercises on series and sequences. Where do I go about starting this question? Any hints and solutions would be most appreciated.

Thank you

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  • $\begingroup$ Using properties of summation, $\sum a_n+b_n=\sum a_n+\sum b_n$ and since each of those series are convergent, their sum is convergent. $\endgroup$ – aleden Nov 5 '17 at 22:40
  • $\begingroup$ Start with the definition. What does it mean for $\sum a_n$ to converge? $\endgroup$ – John Griffin Nov 5 '17 at 22:42
  • $\begingroup$ when one's lost, one usually goes back to the definitions of the concepts. $\endgroup$ – Gabriel Romon Nov 5 '17 at 22:46
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The first step is to go back to the definitions. A series $\sum_{n=1}^{\infty}a_n$ is convergent if $\lim_{N\rightarrow\infty}\sum_{n=1}^{N}a_n$ exists.

Therefore, define $(s_k)$ to be the sequence where $s_k = \sum_{n=1}^{k}a_n$, and $(t_k)$ to be the sequence where $t_k = \sum_{n=1}^{k}b_n$. Now you have that $\sum_{n=1}^{\infty}a_n = \lim_{n\rightarrow\infty} s_n$, and $\sum_{n=1}^{\infty}b_n = \lim_{n\rightarrow\infty} t_n$, so you can do the rest with limit laws.

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