How Can We Tell If a Series Converges or Diverges?

In 12th grade pre-calculus, it’s really important to understand series. This means knowing the difference between finite and infinite series and figuring out if they converge or diverge.

Definitions:

• Finite Series:
  This is a sum with a limited number of terms. For example, if we have $S_n = a_1 + a_2 + ... + a_n$, it has $n$ terms. Since $n$ is a specific number, a finite series always converges to a certain value.

• Infinite Series:
  This is a sum with an infinite number of terms, written as $S = a_1 + a_2 + a_3 + ...$. A big question to ask about infinite series is whether they converge to a specific number or diverge.

How to Test for Convergence:

1. Nth-Term Test for Divergence:
   If the limit of the series $a_n$ as $n$ goes to infinity is not equal to 0, then the series $\sum_{n=1}^{\infty} a_n$ diverges. If it equals 0, we can’t tell for sure.

2. Geometric Series Test:
   A geometric series looks like this: $S = a + ar + ar^2 + ...$. It converges if the absolute value of the common ratio $|r| < 1$. The sum can be found using the formula $S = \frac{a}{1 - r}$.

3. P-Series Test:
   A series like $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

4. Comparison Test:
   For two series $\sum a_n$ and $\sum b_n$, if $0 \leq a_n \leq b_n$ for all $n$, and $\sum b_n$ converges, then $\sum a_n$ also converges.

5. Ratio Test:
   For the series $\sum a_n$, find the limit $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$:

   • If $L < 1$, the series converges.
   • If $L > 1$, the series diverges.
   • If $L = 1$, we can't decide.

By using these tests, you can figure out if a series converges or diverges. This will help you improve your math skills as you study calculus and advanced topics!