SOLUTION: Find the range of values of x for which the infinite series 1 + lnx/2 +(lnx)^2/(2^2) + (lnx)^3/(2^3) + (lnx)^4/(2^4)+... converges. Find the sum to infinity when x=e^(1/2).

Algebra ->  Sequences-and-series -> SOLUTION: Find the range of values of x for which the infinite series 1 + lnx/2 +(lnx)^2/(2^2) + (lnx)^3/(2^3) + (lnx)^4/(2^4)+... converges. Find the sum to infinity when x=e^(1/2).      Log On


   

Question 1069243: Find the range of values of x for which the infinite series 1 +
lnx/2 +(lnx)^2/(2^2) + (lnx)^3/(2^3) + (lnx)^4/(2^4)+... converges. Find the sum to infinity when x=e^(1/2).
Answer by trsomas23@gmail.com(17) About Me  (Show Source):
You can put this solution on YOUR website!
It is an infinite geometric series with common ratio
r = ln%28x%29%2F2
The series converges if
|r| < 1
ln%28x%29%2F2 < 1
ln(x) < 2
x < e%5E2
Sum of series = a%2F%281-r%29
= 1%2F%281-ln%28x%29%2F2%29
When x = e%5E%281%2F2%29, then the sum of the series
= 1%2F%281-1%2F4%29
= 1%2F%283%2F4%29
= 4%2F3