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حدود الـseries  تكون موجبة وسالبة وتكون على الشكل

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots=\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n}$

او

$-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\frac{4}{5}-\frac{5}{6}+\frac{6}{7}-\dots=\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+1}$

والحد النونى يكون على الشكل

$a_{n}=(-1)^{n-1} b_{n} \quad$ or $\quad a_{n}=(-1)^{n} b_{n}$

Alternating serles Test If the alternating series

$\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}=b_{1}-b_{2}+b_{3}-b_{4}+b_{5}-b_{6}+\dots \quad b_{n}>0$

satisfies (i) $b_{n+1} \leq b_{n}$  for all $n$

(ii) $\lim _{n \rightarrow \infty} b_{n}=0$

then the series is convergent.

نظرية:

Alternating Series Estimation Theorem If $s=\Sigma(-1)^{n-1} b_{n},$ where $b_{n}>0,$ is

the sum of an alternating series that satisfies

(i) $b_{n+1} \leq b_{n}$  and (ii) $\lim _{n \rightarrow \infty} b_{n}=0$      $\;|R_{n}|=|s-s_{n}| \leq b_{n+1}$

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