Use the power series for 1 1−x to find a power series representation of f(x) = ln(1−x). what is the radius of convergence? (note: you don’t need to use the ratio test here because we know the radius of convergence of the series p∞ n=0 x n .) (b) use part (a) to find a power series for f(x) = x ln(1 − x). (c) by putting x = 1 2 in your result from part (a), express ln 2 as the sum of an infinite series

The power series of f(x) = ln(1 - x) is  , the power series of xln(1 - x) is   and the value of ln(2) is .

Given :

f(x) = ln (1−x)

a) The integration of 1/(1 - x) is given by:

When |x| >1 :

Now, integrate on both sides in the above equation.

  --- (1)

Now, at (x = 0) the above expression becomes:

By simplifying the above expression in order to get the value of C.

C = 0

Now, substitute the value of C in expression (1).

Now, by shifting the index the above expression becomes:

b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).

c) Now, substitute the value x = 1/2 in the expression (1).

For more information, refer to the link given below:

link

a. Recall that

For , we have

By integrating both sides, we get

If , then

so that

We can shift the index to simplify the sum slightly.

b. The power series for can be obtained simply by multiplying both sides of the series above by .

c. We have

b

Step-by-step explanation: