Convergence of Sequences and Series

Exam Tip: If you encounter a limit involving $\sin(n)$ or $\cos(n)$, which oscillate and have no limit, try using the Squeeze Theorem. Since $-1 \leq \sin(n) \leq 1$, you can often bound the entire expression to force a limit of zero.

From Sequences to Series

While a sequence is a list, a series $\sum_{n=1}^{\infty} a_n$ is the sum of the terms in a sequence. To determine if an infinite sum actually equals a finite number, we look at the partial sums $S_N = \sum_{n=1}^{N} a_n$. A series converges to a sum $S$ if the sequence of these partial sums ${S_N}$ converges to $S$.

The Divergence Test

Before attempting to sum a series, always perform the "Divergence Test." If the limit of the individual terms is not zero ($\lim_{n\to\infty} a_n \neq 0$), then the series must diverge.

Warning: The converse is not true! If $\lim_{n\to\infty} a_n = 0$, the series might still diverge (like the harmonic series $\sum \frac{1}{n}$). The test is only useful for proving divergence, never for proving convergence.

Specialized Convergent Series

Certain types of series follow predictable patterns that allow us to find their exact sums easily.

Geometric Series

A geometric series takes the form $\sum_{n=0}^{\infty} ar^n$, where $r$ is the common ratio.

• It converges if and only if $|r| < 1$.
• The sum is given by $S = \frac{a}{1-r}$, where $a$ is the first term.

For example, the series $\sum_{n=0}^{\infty} (\frac{2}{3})^n$ converges because $|\frac{2}{3}| < 1$, resulting in a sum of $\frac{1}{1 - 2/3} = 3$.

Telescoping Series

A telescoping series is one where the middle terms cancel out when you write out the partial sums, leaving only a few terms at the beginning and the end. A classic example is: $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ By using partial fraction decomposition, we can rewrite the term as: $\frac{1}{n} - \frac{1}{n+1}$ When we sum these, the $-\frac{1}{2}$ from the first term cancels the $+\frac{1}{2}$ from the second, and so on, eventually leaving us with $1 - \lim_{N\to\infty} \frac{1}{N+1} = 1$.

Convergence Tests for General Series

When a series is neither geometric nor telescoping, we use various tests to determine its behavior:

1. p-Series Test: The series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.
2. Comparison Tests:
   • Direct Comparison: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ also converges.
   • Limit Comparison: If $\lim_{n\to\infty} \frac{a_n}{b_n} = L$ (where $0 < L < \infty$), then both series behave the same way.
3. Ratio Test (d'Alembert): Useful for factorials and powers. Let $L = \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges absolutely; if $L > 1$, it diverges.
4. Root Test (Cauchy): Useful for terms raised to the $n$-th power. Let $L = \lim_{n\to\infty} \sqrt[n]{|a_n|}$. The convergence criteria are the same as the Ratio Test.
5. Integral Test: If $f(x)$ is a continuous, positive, decreasing function such that $a_n = f(n)$, then $\sum a_n$ converges if and only if the integral $\int_{N}^{\infty} f(x) \, dx$ converges.