Sum of Geometric Sequence/Examples

Examples of Sum of Geometric Sequence

$\dfrac 1 7$ from $1$ to $n$

$\ds \sum_{j \mathop = 0}^n \dfrac 1 {7^j} = \frac 7 6 \paren {1 - \frac 1 {7^{n + 1} } }$

Common Ratio $1$

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:

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

Index to $-1$

Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$.

Then the formula for Sum of Geometric Sequence:

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

still holds when $n = -1$:

$\ds \sum_{j \mathop = 0}^{-1} x^j = \frac {x^0 - 1} {x - 1}$

Index to $-2$

Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$.

Then the formula for Sum of Geometric Sequence:

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

breaks down when $n = -2$:

$\ds \sum_{j \mathop = 0}^{-2} x^j \ne \frac {x^{-1} - 1} {x - 1}$