Find out the answer to "Determine whether the geometric series is convergent or..."

vazelinahS

Answered question

2021-11-02

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

10 - 2 + 0.4 - 0.008 + \dots

Answer & Explanation

unessodopunsep

Skilled2021-11-03Added 105 answers

Geometric series:
$\sum_{n = 0}^{\infty} a \cdot r^{n} = \frac{a}{1 - r}$
Where r and a are constants
If |r|<1, then the series converges to $\frac{a}{1 - r}$
$10 - 2 + 0.4 - 0.08 \dots$
$10 + 10 \cdot (- \frac{1}{5}) + 10 \cdot {(- \frac{1}{5})}^{2} + 10 \cdot {(- \frac{1}{5})}^{3} \dots$
$\sum_{n = 0}^{\infty} 10 \cdot {(- \frac{1}{5})}^{n}$
This is a geometric series with common ration $r = - \frac{1}{5}$ and Initial Term $a = 10$
Since $| r | = \frac{1}{5} < 1$ , the given geometric series converges.
Sum of the geometric series is
$S = \frac{a}{1 - r} = \frac{10}{1 - (- \frac{1}{5})} = \frac{10}{1 + \frac{1}{5}} = \frac{10}{\frac{6}{5}} = \frac{50}{6} = \frac{25}{3}$
The series converges to $\frac{25}{3}$

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