Find the sum of the following series.
$1 + (1 + a) + r + (1 + a + a^{2}) r^{2} + . . . .$ to $\infty$ $(| r | < 1)$ .

Open in App

Solution

$S = 1 + (1 + a) r + (1 + a + a^{2}) r^{2} + . . . . . . . \infty$
$a r S = a r + (a + a^{2}) r^{2} + (a + a^{2} + a^{3}) r^{3} + . . . . . . . \infty$
$(1 - a r) S = 1 + r + r^{2} + r^{3} + . . . . . . \infty$
$(1 - a r) S = \frac{1}{1 - r}$
$S = \frac{1}{(1 - r) (1 - a r)}$

Suggest Corrections

Similar questions

\frac{1}{⌊ r} + \frac{⌊ 2}{⌊ r + 1} + \frac{⌊ 3}{⌊ r + 2} + . . .

\frac{p^{r}}{⌊ r} + \frac{p^{r - 1}}{⌊ r - 1} \cdot \frac{q}{1} + \frac{p^{r - 2}}{⌊ r - 2} \cdot \frac{q^{2}}{⌊ 2} + \frac{p^{r - 3}}{⌊ r - 3} \cdot \frac{q^{3}}{⌊ 3} + . . . . .

r + 1

\sum_{r = 1}^{n} (- 1)^{r - 1} .^{n} C_{r} (a - r)

\sum_{r = 1}^{n} r (r + 1) (r + 2) (r + 3)

a + (a + d) r + (a + 2 d) r^{2} + . . . . (a + ¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ n - 1 . d) r^{n - 1}

r < 1

Join BYJU'S Learning Program