Infinite Sequences and Series Glossary

Series whose partial sums approach a finite limit.

Series whose partial sums do not approach a finite limit.

\(\sum ar^n\): converges to \(a/(1-r)\) if \(|r|<1\).

All \(x\) where power series converges, including endpoint checks.

\(R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\) for some \(c\) between \(a\) and \(x\).

Taylor series centered at \(a=0\).

\(S_N=\sum_{n=1}^{N}a_n\). Converges if \(\lim S_N\) exists.

\(\sum c_n(x-a)^n\): function as infinite polynomial.

\(R\): power series converges when \(|x-a|<R\).

An ordered list \(a_1,a_2,a_3,\ldots\) defined by a rule.

Sum of sequence terms: \(\sum_{n=1}^{\infty} a_n\).

\(T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k\).