Question

Write out the first few terms of the geometric​ series, Summation from n equals 0 to infinity (StartFraction x minus 4 Over 4 EndFraction )Superscript n ​, to find a and​ r, and find the sum of the series.​ Then, express the​ inequality, StartAbsoluteValue r EndAbsoluteValue less than 1​, in terms of x and find the values of x for which the inequality holds and the series converges.

Answer 1

Answer:

a=1 converges for 0<x<8

Step-by-step explanation:

Given that a geometric series infinite is given in summation form as

$\Sigma _0^\infty (\frac{x-4}{4} )^n$

Here I term is when n =0

So a= I term = $(\frac{x-4}{4} )^0=1$

r = common ratio = $\frac{x-4}{4}$

This series converges only if

$|r|$

$|\frac{x-4}{4}|$

For 0<x<8, we get absolute value of r is less than 1.

Hence series converges for $0$