Need Help?

  • Notes
  • Comments & Questions

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

No comments yet

Join the conversation

Join Notatee Today!