Infinite series Knowledge-Based Study

Study notes

1. What is an Infinite Series?

   An infinite series is the formal sum of an infinite sequence of terms: $$\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots$$

   Sequence of Partial Sums

   To evaluate whether an infinite series adds up to a finite real number, we define the n-th partial sum, denoted by $S_n$:

   $$S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$$

   We say that the series $\sum_{n=1}^{\infty} a_n$ converges to the sum $S$ if the sequence of partial sums $\{S_n\}$ has a finite limit as $n\to\infty$:

   $$\lim_{n \to \infty} S_n = S$$

   If $\lim_{n \to \infty} S_n$ does not exist or equals $\pm\infty$, the series diverges.

   The Divergence Test (n-th Term Test)

   If $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n \to \infty} a_n = 0$. Consequently, its contrapositive provides a practical preliminary test:

   Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$ or does not exist, then the series $\sum_{n=1}^{\infty} a_n$ diverges.

   Caution: If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive. For example, the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges even though $\lim_{n \to \infty} \frac{1}{n} = 0$.

   Geometric Series

   A geometric series has the form: $$\sum_{n=0}^{\infty} a r^n = a + ar + ar^2 + \cdots$$

   • If $|r| < 1$, the series converges to $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.
   • If $|r| \ge 1$, the series diverges.

   The Integral Test and p-Series

   Suppose $f(x)$ is a continuous, positive, and decreasing function on $[1, \infty)$ such that $f(n) = a_n$. Then $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x)\,dx$ either both converge or both diverge.

   Applying the Integral Test to $f(x) = \frac{1}{x^p}$ yields the p-Series Test: $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

   • Converges if $p > 1$.
   • Diverges if $p \le 1$.

   Direct and Limit Comparison Tests

   For series with positive terms $a_n > 0$ and $b_n > 0$:

   • Direct Comparison Test: If $a_n \le b_n$ for all large $n$, and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n \ge b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
   • Limit Comparison Test: If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

   Alternating Series Test (Leibniz's Test)

   An alternating series has the form $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ with $b_n > 0$. The series converges if both conditions hold:

   1. $b_{n+1} \le b_n$ for all $n$ (the terms are non-increasing).
   2. $\lim_{n \to \infty} b_n = 0$.

   Alternating Series Estimation Theorem

   If an alternating series satisfies the test and sums to $S$, the error in approximating $S$ with the $n$-th partial sum $S_n$ satisfies: $$|R_n| = |S - S_n| \le b_{n+1}$$

   Absolute vs. Conditional Convergence

   • A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges. Absolute convergence implies ordinary convergence.
   • A series is conditionally convergent if $\sum a_n$ converges, but $\sum |a_n|$ diverges (e.g., the alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$).

   The Ratio Test

   Let $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.

   • If $L < 1$, the series $\sum a_n$ is absolutely convergent.
   • If $L > 1$ or $L = \infty$, the series diverges.
   • If $L = 1$, the test is inconclusive (use another test such as Comparison or Integral Test).

   The Ratio Test is particularly effective for series involving factorials (e.g., $n!$) or exponential expressions (e.g., $2^n$).

   The Root Test

   Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$.

   • If $L < 1$, the series converges absolutely.
   • If $L > 1$ or $L = \infty$, the series diverges.
   • If $L = 1$, the test is inconclusive.

   The Root Test is useful when terms involve expressions raised to the $n$-th power, such as $(a_n)^n$.