Question

The following series are geometric series or a sum of two geometric series. Determine whether each series converges or not. For the series which converge, enter the sum of the series. For the series which diverges enter "DIV" (without quotes).
a. Σ[infinity] n=1 6^n/5^n = _________
b. Σ[infinity] n=2 1/2^n = ________
c. Σ[infinity] n=5 5^n/6^n = _______

Answer 1

Answer:

a. Does not converge

b. Converges. And the sum would be 0.5

c. Converges. And the sum would be 2.411

Step-by-step explanation:

Remember that a series of the type

$\sum_{n = a}^{\infty} r^n = \frac{r^a}{1-r}$

Converges when  

$|x|<1$

a.

$|\frac{6}{5}| = 1.2 > 1$

Therefore it does not converge.

b.  

$|\frac{1}{2}| = 0.5 < 1$

Therefore it converges. And the sum would be

$\frac{(1/2)^{2}}{1-(1/2)} = \frac{1}{2}$

c.

$|\frac{5}{6}| = 0.833 < 1$

Therefore it converges. And the sum would be

$\frac{(5/6)^{5}}{1-(5/6)} = \frac{3125}{1296} = 2.411$