Sum of Geometric Sequence/Examples/Common Ratio 1

Theorem
Consider the Sum of Geometric Progression defined on the standard number fields for all $x \ne 1$.


 * $\displaystyle \sum_{j \mathop = 0}^n a x^j = a \left({\frac {1 - x^{n + 1} } {1 - x} }\right)$

When $x = 1$, the formula reduces to:


 * $\displaystyle \sum_{j \mathop = 0}^n a 1^j = a \left({n + 1}\right)$

Proof
When $x = 1$, the is undefined:
 * $a \left({\dfrac {1 - 1^{n + 1} } {1 - 1} }\right) = a \dfrac 0 0$

However, the degenerates to: