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

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

  • $\begingroup$ 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

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.