19 - Absolute Convergence implies Convergence

Absolute Convergence implies Convergence

Let $(a_k)$ be a sequence

If ${\sum }_{k=1}^{\infty }|a_k|$ converges then

$$\sum _{k=1}^{\infty }{a}_k\text{ converges}$$

Proof

Let

$$s_n=\sum _{k=1}^n{a}_k\text{ and }{S}_n=\sum _{k=1}^n|a_k|$$

For $n>m$ then

$$|s_n-s_m|=\left|\sum _{k=m+1}^n{a}_k\right|\le \sum _{k=m+1}^n|a_k|=|S_n-S_m|$$

As $\sum |a_k|$ converges by assumption
Then $(S_n)$ is Cauchy by Cauchy Convergence Criterion
So $(s_n)$ is Cauchy by the inequality above
Hence $\sum a_k$ converges by the Cauchy Convergence Criterion