MATHEMATICAL ANALYSIS 12
Infinite Series and Convergence
An infinite series is understood through the convergence of its finite partial sums.
Partial sums
For a sequence \((a_n)\), define
\[S_N=\sum_{n=1}^{N}a_n.\]The infinite series \(\sum_{n=1}^{\infty}a_n\) converges to \(S\) precisely when \(S_N\to S\).
Geometric series
For \(|r|\lt1\),
A necessary condition
If \(\sum a_n\) converges, then \(a_n\to0\). The converse is false: the harmonic series \(\sum1/n\) diverges.
Absolute convergence
If \(\sum |a_n|\) converges, then \(\sum a_n\) converges. This is called absolute convergence.
Common convergence tests
Comparison, ratio, root and integral tests provide different ways to establish convergence. No single test is best for every series.
Connection to Taylor series
A Taylor series is an infinite series of powers. Taylor's theorem gives finite polynomial approximations; additional convergence analysis is needed before identifying a function with its infinite Taylor series.