Sum of Geometric Sequence/Examples/Common Ratio 1

Theorem

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

$\ds \sum_{j \mathop = 0}^n a x^j = a \paren {\frac {1 - x^{n + 1} } {1 - x} }$

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

$\ds \sum_{j \mathop = 0}^n a 1^j = a \paren {n + 1}$

Proof

When $x = 1$, the right hand side is undefined:

$a \paren {\dfrac {1 - 1^{n + 1} } {1 - 1} } = a \dfrac 0 0$

However, the left hand side degenerates to:

 $\ds \sum_{j \mathop = 0}^n a 1^j$ $=$ $\ds \sum_{j \mathop = 0}^n a$ $\ds$ $=$ $\ds a \paren {n + 1}$

$\blacksquare$