# Question about convergence of a series - need help understanding a proof for the strong law of large numbers

My question:

• In the proof below for the strong LLNs, I don't know how the author goes from the $$\color{red}{\text{red box}}$$ to the $$\color{blue}{\text{blue box}}$$ (see screenshot below)

• Getting to the red box makes sense to me. The red box says that each term in the blue box is finite. Note that the very beginning of the proof states $$K = E[X_i^4] < \infty$$

• But just like $$1+1/2+1/3+...$$ goes to infinity, how does showing every term of the sequence is finite prove that the series converges?

• I forgot all this stuff from HS Calculus and could use help, thank you.

Proof from the book:

• The series $\sum\frac1{n^2}$ is convergent. – Lord Shark the Unknown Oct 5 '18 at 4:04
• @LordSharktheUnknown Thanks for your comment, that helps! Hmmm what trips me up is the numerator. The series $$E\left[\frac{X_1^4}{1^4}\right]+E\left[\frac{(X_1+X_2)^4}{2^4}\right]+...$$ looks like it would converge, but how do you know the numerator doesn't mess with $\sum \frac{1}{n^2}$? Thanks – HJ_beginner Oct 5 '18 at 4:07
• @LordSharktheUnknown ahhh I think I see it now, instead of thinking about it like $$E\left[\frac{X_1^4}{1^4}\right]+E\left[\frac{(X_1+X_2)^4}{2^4}\right]+...$$ I should think about it as $$E \left[ \sum_{n=1}^\infty \frac{S_n^4}{n^4}\right] \le \sum_{n=1}^\infty \frac{K}{n^3} + \frac{3K}{n^2}$$ which converges... is that correct? – HJ_beginner Oct 5 '18 at 4:10
• Yes and that's why we do some calculations to find the bound before. – BGM Oct 5 '18 at 5:25
• @BGM Thanks for your help and time! – HJ_beginner Oct 5 '18 at 5:57