Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
7 + 6 + 36/7 + 216/49 + ...
If it is convergent, find its sum.

Answer 1

Answer:

- L < 1, so by series ratio test, It is convergent

- sum of the convergent series is 49

Step-by-step explanation:

Given the data in the question;

series = 7 + 6 + 36/7 + 216/49 + .........

⇒ 7 + 7 × 6/7 + 7 × 36/49 + 7 × 216/343 +........

⇒ 7 + 7 × 6/7 + 7 × 6²/7² + 7 × 6³/7³ .....

⇒ 7( 1 + 6/7 + 6²/7² + 6³/7³ + ..... )

⇒ 7 ∞∑_$_{n=0$ $($ 6/7 $)^n$

⇒ ∞∑_$_{n=0$ ( $6^n$/$7^{n-1$ )

So, L =  $\lim_{n \to \infty} | \frac{a_{n} + 1}{a_n} |$

= $\lim_{n \to \infty} | \frac{\frac{6^n+1}{7n} }{\frac{6^n}{7^n-1} } |$

= $\lim_{n \to \infty} | \frac{6}{7} |$

= 6 /7

L < 1

so by series ratio test,

It is convergent

So we find the sun;

Sum of  infinite geometric series is;

⇒ a / (1-r)

here, a = first number and r = common ratio

∑ = 7( 1 + 6/7 + 6²/7² + 6³/7³ + ..... )

a = 1 and r = 6/7

so

∑ = 7( $\frac{1}{1 - \frac{6}{7} }$ )

= 7( $\frac{1}{\frac{7-6}{7} }$ )

= 7( $\frac{1}{\frac{1}{7} }$ )

= 7( 7 )

= 49

Therefore, sum of the convergent series is 49