Numerical Series

A numerical series is the sum of infinitely many terms: $\sum_{n=0}^\infty a_n$. The partial sum $S_N = \sum_{n=0}^N a_n$ is the sum of the first $N+1$ terms. The series converges to $S$ if $\lim_{N\to\infty} S_N = S$ (exists and is finite). If the limit of partial sums does not exist or is infinite, the series diverges.

Necessary (but not sufficient) condition for convergence: $a_n \to 0$. Warning: it is not sufficient! The harmonic series $\sum_{n=1}^\infty 1/n$ diverges even though $1/n\to0$.

Geometric series: $\displaystyle\sum_{n=0}^\infty q^n = \frac{1}{1-q}$ for $|q|<1$; diverges for $|q|\geq1$. The partial sum is $S_N = (1-q^{N+1})/(1-q)$.

Riemann $p$-series: $\displaystyle\sum_{n=1}^\infty \frac{1}{n^\alpha}$ converges if and only if $\alpha > 1$. Proof via the integral test: $\int_1^\infty x^{-\alpha}dx$ converges iff $\alpha>1$.

Absolute convergence: if $\sum|a_n|$ converges, then $\sum a_n$ converges (absolutely). Absolute convergence implies ordinary convergence, but not vice versa (e.g., alternating harmonic series converges conditionally but not absolutely).

Ratio test (D'Alembert): $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$. Then:

• $L<1$: series converges absolutely
• $L>1$ (or $L=+\infty$): series diverges
• $L=1$: test is inconclusive

Especially useful for series with factorials ($n!$) or exponentials ($q^n$).

Root test (Cauchy): $L = \lim_{n\to\infty}\sqrt[n]{|a_n|}$. Same three conclusions as the ratio test. Often easier when $a_n$ involves $n$-th powers.

Comparison test: if $0\leq a_n\leq b_n$ for all sufficiently large $n$, then: $\sum b_n$ converges $\Rightarrow\sum a_n$ converges; $\sum a_n$ diverges $\Rightarrow\sum b_n$ diverges.

Limit comparison test: if $a_n,b_n>0$ and $\lim a_n/b_n = L\in(0,+\infty)$, then $\sum a_n$ and $\sum b_n$ have the same behaviour (both converge or both diverge).

Leibniz alternating series test: $\sum_{n=0}^\infty (-1)^n b_n$ converges if $b_n$ is monotonically decreasing and $b_n\to0$. Convergence is conditional (not absolute if $\sum b_n$ diverges). Moreover $|S-S_N|\leq b_{N+1}$ (error bound).