MATH SOLVE

4 months ago

Q:
# Find the value of b for which 1+e^b+e^2b+e^3b+...=9

Accepted Solution

A:

this is a geometric series, let r=e^b then we have

1+r+r^2+r^3+....=9

1/(1-r)=9

1-r=(1/9)

r=1-(1/9)

r=8/9

e^b=8/9 take ln of both sides

ln(e^b)=8/9 ln(x^y)=y*ln(x) and ln(e)=1 so

b*ln(e)=b=ln(8/9)

thus

b=ln(8/9)

1+r+r^2+r^3+....=9

1/(1-r)=9

1-r=(1/9)

r=1-(1/9)

r=8/9

e^b=8/9 take ln of both sides

ln(e^b)=8/9 ln(x^y)=y*ln(x) and ln(e)=1 so

b*ln(e)=b=ln(8/9)

thus

b=ln(8/9)