Lecture 44: Infinite Series

  • {an{a_n}} is an infinite sequences \rightarrow a1+a2+a3a_1 + a_2 +a_3  ... n=1an\sum_{n=1}^{\infty}a_n 
  • The nthn^{th} partial sum (sn)(s_n) is sn=a1+a2...ans_n = a_1 + a_2 ... a_n 
  • If limnsn=s\lim_{n\rightarrow \infty} s_n =s  therefore infinite series is convergent where ss = sum
  • n=1an\sum_{n=1}^{\infty} a_n diverges if limn0an0\lim_{n\rightarrow 0}a_n \neq 0  or it fails to exist


Divergence Test

  • If limn0an0\lim_{n\rightarrow 0}a_n \neq 0  or DNE then n=1an\sum_{n=1}^{\infty} a_n is divergent

  • n=1arn1=a1r \sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}
  • Converges to this if r<1|r| < 1 whereas if r1|r| \geq 1 series diverges


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