using appropriate taylor series to find the first four nonzero terms of an infinite series that equal to ln(4/3).

Answer 1

To find the terms of an infinite series that equals ln(4/3), we can use the Taylor series expansion of the natural logarithm function, ln(x), centered at x = 1.

The Taylor series expansion of ln(x) is given by:

ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + ...

Since we want to find the terms that equal ln(4/3), we substitute x = 4/3 into the Taylor series:

ln(4/3) = (4/3 - 1) - (4/3 - 1)^2/2 + (4/3 - 1)^3/3 - (4/3 - 1)^4/4 + ...

Simplifying this expression, we have:

ln(4/3) = (1/3) - (1/9)/2 + (1/27)/3 - (1/81)/4 + ...

Now, we can see that the first four terms of this infinite series are:

1/3, -(1/18), 1/81, -(1/324)

So, the first four nonzero terms of the infinite series that equals ln(4/3) are 1/3, -(1/18), 1/81, and -(1/324).