log(2)+log(2/3)+log(2/3^2)+log(2/3^3)+.... The sum of the first 10 terms equals?

1 Answer
Jan 11, 2018

10*ln(2)-45ln(3)~~-42.5

Explanation:

Use the logarithm rule ln(a)+ln(b)=ln(ab)

f_1=ln(2)=ln(2/3^0)

f_2=ln(2)+ln(2/3)=ln(2^2/3^1)

f_3=ln(2)+ln(2/3)+ln(2/3^2)=ln(2^3/3^3)

f_4=ln(2)+ln(2/3)+ln(2/3^2)+ln(2/3^3)=ln(2^4/3^6)

A pattern occurs

f_n=ln(2^n/3^(((n^2-n)/2)))=nln(2)-(n^2-n)/2ln(3)

For n=10

f_10=10*ln(2)-(10^2-10)/2ln(3)

f_10=10*ln(2)-45ln(3)

f_10~~-42.5