The geometric series of $\sum _{n=0}^\infin a\cdot r^n$ or $\sum _{n=0}^\infin a\cdot r^{n-1}$

• Diverges if $|r|\ge 1$.

  • Since $a_\infin$ is infinity, therefore the $S_n$ is equal to definitely infinity. (diverges)

• Converges to $\frac {a_1} {1-r}$

  <aside> 📌 $a_1$ is not a, it means to the first term of the series. This applied to sum where $n\ne 0$

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• if $a_\infin$ is a finite number, then $S_n$ is definitely finite. (converge)

• the formula for a geometric series is $S_n=a_1(\frac {1-r^n} {1-r})$ and since $r_\infin$ is approaching to a very small number (zero), the formula of a infinite geometric series is $S_\infin=\frac {a_1} {1-r}$