1. This series converges. Do fraction subtraction to see that it is equal to . You can do limit comparison of this series with .
2. This series also converges. You can compare it (not limit comparison) to .
3. This series diverges. Let . If we also let , then we have . By using the comparison test, we can say diverges when diverges.
So now we just need to check that diverges. For this, we can just use limit comparison with . This is a slightly different explanation than that given in class. If you understood the in-class version better, feel free to stick with it. Either type of answer will be acceptable on the midterm.
4. This problem is written incorrectly since the 's should have started with 2 instead of 1. By starting at 1, there is a division by zero in the first term. This is what lead to the confusion about how to apply the integral test in class.
Let me review what I said about the integral test in class. If you know is a positive and decreasing function, then you can always use the test to see if diverges or converges. Usually you try to evaluate the integral
Anyway, let's do the integration. We'll use 2 as the lower limit of integration. Make the substitution so that , and the integral becomes
5. Use the ratio test to find the interval of convergence.
6. We already know that
7. Start with
8. Start with the Taylor series for , which is
9. For this one, you'll have to use the Taylor series theorem, which says that if a function has any power series representation at all (some functions have none), then it is given by:
10. First we need the series expansion for , which is