Chapter 5

Previous: Chapter 4

Sequences

1. \(\lim\limits_{n\rightarrow \infty} a^{\frac{1}{n}}= 1\)
2. \(\lim\limits_{n\rightarrow \infty} n^{\frac{1}{n}}= 1\)
3. \(\lim\limits_{n\rightarrow \infty} r^n= 0\) for \(-1 < r <1\)
4. \(\lim\limits_{n\rightarrow \infty} \left(1 + \frac{a}{n} \right)^{\frac{1}{n}}= e^{a}\)
   \(\sum^{\infty}_{k=1}ar^{k-1}\) converges if \(-1 < r <1\)
   \(\sum^{\infty}_{k=1} ar^{k-1}=\frac{a}{1-r}\) for \(-1<r<1\)
   \(\sum^{\infty}_{k=1} ar^{k-1}=\frac{1}{kp}\) The series is convergent if \(p>1\) and is divergent if \(p\leq 1\)

Ratio Test

Given a power series
\(\sum\limits^{\infty}_{k=1}c_k(x-a)^{k}\)
let \(a_{k}=c_k(x-a)^k\)

let \(L=\lim\limits_{k \rightarrow \infty} |\frac{a_{k}+1}{a_k}|\)
if \(L < 1\) the power series converges

Root Test

Let \(L = \lim\limits_{k \rightarrow \infty}|a_k^\frac{1}{k}|\)
if \(L < 1\) (including \(L=0\)), the power series converges
if \(L > 1\) (including \(L=\infty\)), the power series diverges

Taylor series

\(\sum\limits^{\infty}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k} =\sum\limits^{\infty}_{k=0}\frac{e^a}{k!}(x-a)^k\)

Maclaurin series

\(\sum\limits^{\infty}_{k=0}\frac{f^{(k)}(0)}{k!}(x)^{k}\)

Taylor polynomial

Given a function \(f\), its polynomial centered at \(a\), denoted by \(P_n\), is the sum of the first \((n+1)\) terms of its Taylor series \(\sum\limits^{\infty}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}\)
that is \(P_{n}=\sum\limits^{n}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}\)

End of Module