Infinite Sequences and Series Cheat Sheet

The core ideas of Infinite Sequences and Series distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

sum a*r^n converges to a/(1-r) when |r|<1.

sum 1/n^p converges iff p>1.

L = lim |a_{n+1}/a_n|. L<1: converges. L>1: diverges.

f(x) = sum f^(n)(a)/n! (x-a)^n.

Taylor series centered at a=0.

R such that series converges for |x-a|<R.

|R_n(x)| <= M/(n+1)! |x-a|^(n+1).

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

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

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

Convergent Series:Series whose partial sums approach a finite limit.

Divergent Series:Series whose partial sums do not approach a finite limit.

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

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

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

Maclaurin Series:Taylor series centered at \(a=0\).

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

Interval of Convergence:All \(x\) where power series converges, including endpoint checks.

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

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