Convergence

Convergence describes when a sequence or series approaches a finite limit.

Sequence: $\{a_n\}$ converges to $L$ if for every $\varepsilon > 0$ there exists $N$ such that $|a_n - L| < \varepsilon$ for all $n > N$.

Series: $\sum a_n$ converges if its partial sums $S_n$ converge.

Standard tests:

• n-th term test: $a_n \not\to 0$ → diverges.
• Geometric series: $\sum r^n$ converges iff $|r| < 1$.
• Comparison test: bound by a known series.
• Ratio test: $\lim |a_{n+1}/a_n| < 1$ → converges.
• Integral test: connects $\sum a_n$ to $\int_1^\infty f(x) dx$.
• Alternating series test: $\sum (-1)^n b_n$ converges if $b_n \to 0$ monotonically.

Absolute ($\sum |a_n|$ converges) is stronger than conditional convergence. Harmonic series $\sum 1/n$ diverges; $\sum (-1)^n/n$ converges (alternating).