The algebra of convergent series.

Theorem 9.2.1   Let $U=\underset{k=0}{\overset{+\infty }{\sum}} u_n$ and $V=\underset{k=0}{\overset{+\infty }{\sum}} v_n$ be two convergent series. Then the series $\underset{k=0}{\overset{+\infty }{\sum}} (u_n+v_n)$ is convergent and its sum is equal to $U+V$ .

Theorem 9.2.2   Let $U=\underset{k=0}{\overset{+\infty }{\sum}} u_n$ be a convergent series and $\alpha \in \mathbb{R}$ . Then the series $\underset{k=0}{\overset{+\infty }{\sum}} \alpha u_n$ is convergent and its sum is equal to $\alpha U$ .

Together these two theorems mean that the set of convergent series of real numbers is a real vector space.

Remark 9.2.3   Adding or deleting terms from a series does not change the convergence/divergence. For a convergent series, it changes the value of the sum.

Example 9.2.4       

$$\underset{k=2}{\overset{+\infty }{\sum}} \frac {1}{3^n} = \unders... ...\infty }{\sum}} \frac {1}{3^n} - 1 - \frac 13 = \frac 32 - \frac 43 = \frac 16.$$