for .
Why isn't this the case for ? Instead, why do we replace with to get ?
Ok
I will go in depth with this,
We know (hopefully) that
We also know
So now making a substitution we can see that
So now we know that integrating/differentiating a power series only changes the endpoint behavior
So we know that since
converges
So now we must check the endpoints.
At we have
Which diverges by the Alternating series test.
Since
Now we need to check at
We have
Which diverges by the integral test
So we have that
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Now for your second question we can see the by direct substitution into either the log or the equation, or seeing that
This implies that
is found by
Now once again we must check the endpoints,
so at
We have
Which as I showed earlier is divergent by the integral test.
But at
We have
Which as was shown earlier is convergent by the alternating series test.
Hope this helps.