Evaluating an infinite series Write the Taylor series for f(x) = 1n (1 + x) about 0 and find the interval of convergence. Assume the Taylor series converges to / on the interval of convergence. Evaluate /(I) to find the value of (the alternating harmonic series).

Solution 47E

Step 1:

Given that

f(x) = 1n (1 + x) about 0

(the alternating harmonic series).

Step2:

To find

Write the Taylor series for f(x) = 1n (1 + x) about 0 and find the interval of convergence. Assume the Taylor series converges to / on the interval of convergence. Evaluate /(I) to find the value of

Step3:

We have

f(x) = 1n (1 + x)

Taylor serie for f(x) = 1n (1 + x)

In(1+x)=x-

We know that series converges when |x|<1 and diverges when |x|>1

Now we have to check x=

So, the interval of convergence is (-1,1)

The equation

In(x+1)=

Is true for x=1

So, put x=1 we get

In(2)=

That is the sum of the alternating harmonic series is In2.