Sum of Infinite Arithmetic-Geometric Sequence

Theorem

Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as:

$a_k = \paren {a + k d} r^k$ for $n = 0, 1, 2, \ldots$

Let:

$\size r < 1$

where $\size r$ denotes the absolute value of $r$.

Then:

$\ds \sum_{n \mathop = 0}^\infty \paren {a + k d} r^k = \frac a {1 - r} + \frac {r d} {\paren {1 - r}^2}$

Proof

From Sum of Arithmetic-Geometric Sequence, we have:

$\ds s_n = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$

We have that $\size r < 1$.

So by Sequence of Powers of Number less than One:

$r^n \to 0$ as $n \to \infty$

and

$r^{n - 1} \to 0$ as $n - 1 \to \infty$

Hence:

$s_n \to \dfrac a {1 - r} + \dfrac {r d} {\paren {1 - r}^2}$

as $n \to \infty$.

The result follows.

$\blacksquare$

Linguistic Note

In the context of an arithmetic sequence or arithmetic-geometric sequence, the word arithmetic is pronounced with the stress on the first and third syllables: a-rith-me-tic, rather than on the second syllable: a-rith-me-tic.

This is because the word is being used in its adjectival form.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Arithmetic-Geometric Series: $19.7$