SOLUTION: The series v + v^2 + v^3 + . . . has a limiting sum w. a. Write w in terms of v. b. Find v in terms of w. c. Hence find the limiting sum of the series w − w^2 + w^3 − .

Algebra ->  Sequences-and-series -> SOLUTION: The series v + v^2 + v^3 + . . . has a limiting sum w. a. Write w in terms of v. b. Find v in terms of w. c. Hence find the limiting sum of the series w − w^2 + w^3 − .       Log On


   

Question 1175119: The series v + v^2 + v^3 + . . . has a limiting sum w.
a. Write w in terms of v.
b. Find v in terms of w.
c. Hence find the limiting sum of the series w − w^2 + w^3 − . . . , assuming that ∣w∣ < 1.
d. Test your results with v = 1/3
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The series v + v^2 + v^3 + . . . has a limiting sum w.
a. Write w in terms of v.
S%5Binfinity%5D=a%5B1%5D%2F%281-r%29

w=v%2F%281-v%29
b. Find v in terms of w.

w=v%2F%281-v%29
w%281-v%29=v
w-wv=v
w=wv%2Bv
w=v%28w%2B1%29
w%2F%28w%2B1%29=v
v=w%2F%28w%2B1%29

c. Hence find the limiting sum of the series w − w^2 + w^3 − . . . , assuming
that ∣w∣ < 1.
S%5Binfinity%5D=a%5B1%5D%2F%281-r%29=w%2F%281-%28-w%29%29=w%2F%281%2Bw%29
d. Test your results with v = 1/3
w=v%2F%281-v%29
w=%281%2F3%29%2F%281-1%2F3%29
w=%281%2F3%29%2F%282%2F3%29
w=%281%2F3%29%2A%283%2F2%29
w=%281%2Fcross%283%29%29%2A%28cross%283%29%2F2%29
w=1%2F2
------------------------------------------------
v=w%2F%28w%2B1%29
1%2F3=%281%2F2%29%2F%28%281%2F2%29%2B1%29
1%2F3=%281%2F2%29%2F%283%2F2%29
1%2F3=%281%2F2%29%2A%282%2F3%29
1%2F3=%281%2Fcross%282%29%29%2A%28cross%282%29%2F3%29
1%2F3=1%2F3

Edwin