Page 102 - Calculus for the Clueless, Calc II

2. Use the integral test if the infinite series looks like an integral you have done. By this time, you should have
so many integrals you should be familiar with and/or sick of them.

3. Don't use the integral test if you can see an easier one or if there is a factorial symbol.

4. My favorite is the ratio test. Always try the ratio test if there is a factorial or an x in the problem. Also try the
k
k
ratio test if there is something to a power of k, such as 2 or k .
5. Use the limit comparison test or the comparison test if the series looks like one you know like the harmonic
series, p 2 series, and so on. Use the comparison if the algebra is not too bad. Use the limit comparison if the
algebra looks really terrible or even semiterrible.

6. Use the root test if there is at least one term with k in the exponent and no factorial in the problem.

7. If there is a series where there are a lot of messy-looking terms multiplying each other, the ratio test is
probably the correct one.

8. Sometimes you may not be able to tell the terms go to zero. The ratio test may give absolute convergence or
divergence immediately.

9. Practice factorial. It is new to most of you. Once again, note that (2n. 1)! = (2n. 1)(2n)! = (2n + 1)(2n)(2n -
1)!, that is, 7! = 7(6!) = 7(6)(5!). Study factorial!!!!!

10. Most of all, do a lot of series testing. You will get better if you practice. The nice part is that the problems
are mostly very short.

A Preview of Power Series

We would like to have a polynomial approximation of a function in the vicinity of a given point. Polynomials
are very easy to work with. They can be integrated easily, while many functions can't be integrated at all. Exact
answers are usually not needed, since we do not live in a perfect world.

We therefore have Taylor's theorem, which gives us a polynomial that approximates f(x) for every x
approximately equal to a; the closer x is to a, the better the approximation for a given length. The ''meat" of the
theorem is a formula for the remainder, or error, when you replace the function by the polynomial. This is
necessary so that you know how close your answer is.

Taylor's Theorem

1. f( n+1 )(x), [n + 1 derivatives], continuous on some interval, I, where x is in the interval.

2. a is any number in the interval I, usually its midpoint.

3.

Note

S n(x) is the sum of the polynomial terms up to term of degree n.