28. Convergence for Series

Partial Sum of Series ${\sum }_{k=1}^{\infty }{a}_k$

Let $(a_k)$ be a sequence
For $n\ge 1$, let

$$s_n=a_1+a_2+\cdots +a_n=\sum _{k=1}^n{a}_k$$

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Convergence of a Partial Sum

$$\text{Series }\sum _{k=1}^{\infty }\text{ converges if sequence }(s_n)\text{ of partial sum converges}$$

In other words

$$\text{if }{s}_n\to s\text{ as }n\to \infty \text{ then }\sum _{k=1}^{\infty }{a}_k=s$$

If $(s_n)$ does not converge, then

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

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Examples of Series

$$e=\sum _{n=0}^{\infty }\frac{1}{n!}$$ $$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}$$

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Convergence of Series is a special case of convergence for sequences

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Transclude of 24---Series#^52a0cc