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

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

- sum of the convergent series is 49

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) | (a_(n) + 1)/(a_n) |`

=
`\lim_(n \to \infty) | ((6^n+1)/(7n) )/((6^n)/(7^n-1) ) |`

=
`\lim_(n \to \infty) | (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(
`(1)/(1 - (6)/(7) )` )

= 7(
`(1)/((7-6)/(7) )` )

= 7(
`(1)/((1)/(7) )` )

= 7( 7 )

= 49

Therefore, sum of the convergent series is 49