Strategy for Series

Consider a series in the form $\sum_{k=1}^n {u_n}$. To determine whether the series converges or diverges, one of these cases (the most appropriate one) can be used. No need to memorize all these.$$

1. If $\lim_{n\to\infty} u_n \neq 0$ is apparent, then divergence test can be used. Otherwise look out for another way.
2. If $u_n$ consists of a constant raised to a power of $n$, geometric series can be used.
3. If $u_n$ consists of $n$ raised to a constant power, p-series can be used.
4. If $u_n$ consists of $(-1)^n$, alternating series test can be used.
5. If $u_n$ consists of $v_n$ raised to a power of $n$, root test would be suitable.
6. If $u_n$ includes $n!$, ratio test must be used.
7. If $u_n$ is a fraction, and consists of $n$ in both the denominator and the numerator, then direct comparison test or limit comparison test can be used. Consider the dominating parts to choose the $v_n$.
8. If $u(x)$ is a positive and decreasing function, and $\int_a^\infty u(x)\,\text{d}x$ is easy to evaluate, then integral test can be used.

Secret note on inequalities

For any $p\gt0$ and $q>1$, as $n$ tends to $\infty$, the below inequality holds:

$$\ln n \ll n^p \ll q^n \ll n! \ll n^n$$

For $p \le k$: $$

$$n^p \log n \ll n^k$$

For $p \gt k$: $$

$$n^p \log n \gg n^k$$

These inequalities can be used to find corresponding $v_n$ to some $u_n$ to be used for direct comparison test or limit comparison test. The list is found on a video by blackpenredpen on YouTube.