W2 L'Hopital's Rule, Sequences, Limits of Sequences and Series

PRACTICE PRACTICE PRACTICE calculating limits and suing limit laws - no practice = fail - so P R A C T I C E (using problem booklet and textbook)

DIFFERENTIABILITY

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Definition of derivative

• Let $f:\R\rightarrow\R$ be a real-valued function. The derivate of $f$ at $x=a$ is defined by:
  • $\boxed{f'(a)\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}}$
  • The function $f$ is differentiable at $x=a$ is this limit exists
• Geometrically, $f$ is differentiable at $x=a$ if the graph $y=f(a)$ has a tanget line given by:
  • $\boxed{y-f(a)=f'(a)(x-a)}$
  • whihc gives a good approximation to grpah near $x=a$
  • If $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$

L’Hopital’s Rule

• $\boxed{\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)},g'(x)\neq0}$
• if the limit involving the derivatives exists
• L’Hopital’s Rule also holds when $x$ approaches infinity
• apply if $\frac{\infty}{\infty}$ or $\frac{0}{0}$

SEQUENCES

Definition of sequence

• A sequence is a function $f:\N\rightarrow\R$\ $\mathbb{C}$\etc.
• It can be thought of as an ordered list of real numbers
• if you change places of elements, it is a different sequence - because different order

Definition of limit of sequence

• A sequence ${a_n}$ has the limit $L$ if $a_n$ can be made arbitrarily close to $L$ by making $n$ sufficiently large
• $\boxed{\lim_{n\rightarrow\infty}a_n=L}$
• or $a_n\rightarrow L$ as $n\rightarrow\infty$
• If the limit exists we say that the sequence converges.
• Otherwise, we say that the sequences diverges
• If it exists, $L$ must be a unique finite real number

Theorem

Sequences Limit Laws

• Let ${a_n}$ and ${b_n}$ be sequences of real numbers and $c\in\R$ a constant
• If $\lim_{n\rightarrow\infty}a_n$ and $\lim_{n\rightarrow\infty}b_n$

1. $\lim_{n\rightarrow\infty}[a_n+b_n]=\lim_{n\rightarrow\infty}a_n+\lim_{n\rightarrow\infty}b_n$
2. $\lim_{n\rightarrow\infty}[ca_n]=c\lim_{n\rightarrow\infty}a_n$
3. $\lim_{n\rightarrow\infty}[a_nb_n]=\lim_{n\rightarrow\infty}a_n\times\lim_{n\rightarrow\infty}b_n$
4. $\lim_{n\rightarrow\infty}\frac{\lim_{n\rightarrow\infty}a_n}{\lim_{n\rightarrow\infty}b_n}$ provided $\lim_{n\rightarrow\infty}b_n\neq0$;$b_n\neq0$ for sufficiently large $n$
5. $\lim_{n\rightarrow\infty}c=c$

Sequences Sandwich theorem

• Let ${a_n},{b_n}$ and ${c_n}$ be sequences of real numbers
• If $a_n\leq c_n\leq b_n$ for all $n>N$ for some $N$, and
• $\lim_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}b_n=L$
• then
• $\lim_{n\rightarrow\infty}c_n=L$

Standard Limits

#	Standard Limit	Sequence Condition $n$	Function Condition $x$
1	$\lim_{n\rightarrow\infty}\frac{1}{n^p}=0$	$p>0$	$p>0$
2	$\lim_{n\rightarrow\infty}r^n=0$	$	$r<1$ $0\leq r<1$
3	$\lim_{n\rightarrow\infty}a^{\frac{1}{n}}=1$	$a>0$	$a>0$
4	$\lim_{n\rightarrow\infty}n^{\frac{1}{n}}=1$	-	-
5	$\lim_{n\rightarrow\infty}\frac{a^n}{n!}=0$	a\in\R	❌
6	$\lim_{n\rightarrow\infty}\frac{\log n}{n^p}=0$	p>0	p>0
7	$\lim_{n\rightarrow\infty}(1+\frac{a}{n})^n=e^a$	a\in\R	a\in\R
8	$\lim_{n\rightarrow\infty}\frac{n^p}{a^n}=0$	$p\in\R,a>1$	$p\in\R,a>1$

Order of Heirarchy

• The order hierarchy can be used to help identify the largest term in an expression:
• $(\log n)^q\ll n^p\ll a^n\ll (n!)^q\ll e^{\ln^2} ; q,p>0,a>1$
• logarithmic growth $\ll$ algebraic growth $\ll$ exponential growth $\ll$ factorial growth

When to change discrete variable $n$ to real variable $x$

1. before applying L’Hopital’s Rule
2. before applying the continuity theorem

SEQUENCE OF PARTIAL SUMS

• Adding terms of a sequence together:
  • $s_1=a_1$
  • $s_2=a_1+a_2$
  • …
  • $s_n=a_1+a_2+...+a_n$
• The sequence of partial sums ${s_n}$ may or may not converge
• If it does converge: $S=\lim_{n\rightarrow\infty}s_n=\lim{n\rightarrow\infty}(a_1+a_2+...+a_n)$

SERIES

• Series with $a_n$ denoted by the sum:
  • $\sum_{n=1}^\infty a_n$
• IF $\lim_{n\rightarrow\infty}s_n$ exists: Series converges
• OTHERWISE: Series diverges

SEQUENCES VS SERIES

PROPERTIES OF SERIES

• IF $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ converges
• THEN $\sum_{n=1}^\infty (a_n+b_n)$ converges
  • $\sum_{n=1}^\infty (a_n+b_n) = \sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n$
• AND $\sum_{n=1}^\infty (ca_n)$ converges
  • $\sum_{n=1}^\infty (ca_n)=c\sum_{n=1}^\infty a_n$

GEOMETRIC SERIES

• $\sum_{n=1}^\infty ar^n=\frac{a}{1-r}$
• IF $|r|<1$: converges
• IF $|r|\ge1$: diverges

HARMONIC P SERIES

• $\sum_{n=1}^\infty\frac{1}{n^p}$
• IF $p>1$: converges
• IF $p\le1$: diverges

DIVERGENCE TEST

• $\sum_{n=1}^\infty a_n$
• IF $\lim_{n\rightarrow\infty}a_n\ne0$: diverges
• IF $\lim_{n\rightarrow\infty}a_n=0$: inconclusive

COMPARISON TEST

• $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ are positive term series
• IF $a_n\le b_n$ for all $n$ AND $\sum_{n=1}^\infty b_n$ converges: $\sum_{n=1}^\infty a_n$ converges
• IF $a_n\ge b_n$ for all $n$ AND $\sum_{n=1}^\infty b_n$ diverges: $\sum_{n=1}^\infty a_n$ diverges
• compare given series with geometric/harmonic p series
• HOW TO GET $b_n$:
  • find fastest growing terms in numerator and denominator
  • divide numerator and denominator by their respective fastest growing terms
  • see if the result is similar to geometric/harmonic p series
  • Ed answer:
    • 

RATIO TEST

• $\sum_{n=1}^\infty a_n$ & $L=\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}$
• IF $L<1$: converges
• IF $L>1$: diverges
• IF $L=1$: inconclusive
• usually for:
  • exponents
  • factorials