Series

A series is the sum of the terms that form a sequence:

Arithmetic series:

• 1+2+3+.... is an arithmetic series
  • S1=1
  • S2=3
  • S3=6
• $S_n$ also known as the nth partial sum
  • $T_1=S_1=a$
  • $S_n=S_{n-1}+T_n$
  • therefore: $T_n=S_n-S_{n-1}$ $n\ge 2$

finding sums of arithmetic series:

• $S_n=\frac{n}{2}(a+T_n)$
• $S_n=\frac{n}{2}(2a+(n-1)d)$
  • ^ as we don’t always know $T_n$
• Working out:
  
  
  

Geometric series:

sums of geometric series:

$S_{n}=\frac{a(1-r^n)}{1-r}$
or $S_{n}=\frac{a(r^n-1)}{r-1}$
$r\neq 1$

• Working out:
  

sums of infinite geometric series:

A geometric series is:

• Convergent if: $|r|<1$
• Divergent if: $|r|\ge 1$

Convergent geometric series can be added up endlessly, and will still produce a sum:

Rational:

• if $-1<r<1$ (i.e. $|r|<1$)
  Then as $n\to \infty$ , $r^n\to 0$
  • $\therefore S_{\infty }=\frac{a}{1-r},r\ne 1$
  • The sum “converges” to the “limiting sum”