Other Ideas to Focus On

$\lim a_n=0$ does not imply convergence.

Students use the ratio/root test on the endpoints of a power series. It's hard for them to grasp that those are exactly the points where the ratio/root test gives 1, and hence inconclusive. Apply the ratio/root test is a waste of time. Just as bad is when they apply the ratio/root test to the endpoints and get a value other than 1.

$\lim\left(1+\frac{x}{n}\right)^n=e^x$

Stirling's formula.

Variations of common series and how to deal with them. Such as using the limit comparison theorem.

Many students were surprised that for any positive number $b$, $b^{1/n}$ goes to $1$. This is relevant to the root test.

Don't confuse $x$ with $n$ in a power series.

$ln(1)=0$

Unrelated: One problem I thought was interesting was solving $\int f(x) dx \int 1/f(x)dx =-1$. I wrote up a solution that involved making the substitution $f(x)=F^\prime (x)$ or $\int f(x)dx=F(x)$.

Unrelated: $\ln(\tan\theta+\sec\theta)=\frac{1}{2} \ln (1+\sin(\theta))-\ln(1-\sin(\theta))$

7.4.8 u-substitution before finishing. M.K.

Simple thing as finishing the problem. Why stop? Did you integrate all the way?

$-1$ erroneously becomes $(-1)^n$ and similar errors.