theorems of analysis  
 
  L'Hospital's rule  
 
  Integration techniques  
 
  Taylor expansions  
 
  Constants  
 
  home  
 
  ask us  
 

 
      Calculus II

      Contents




© The scientific sentence. 2010

Calculus II: sequences & series
sequences & series calculus





sequences Properties



1. monotonic sequence

A sequence is monotonic if and only if it is either entirely increasing or decreasing.

Given any sequence {an} we have the following:
• We call the sequence increasing if an < an+1 for every n.

• We call the sequence decreasing if an > an+1 for every n.

• If {an} is an increasing sequence or {an} is a decreasing sequence, the sequence is said monotonic.

• If there exists a number m such that m ≤ an for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.

• If there exists a number M such that an ≤ M for every n we say the sequence is bounded above. The number M is sometimes called an upper bound for the sequence.

• If the sequence is both bounded below and bounded above we call the sequence bounded.



2. Limit of sequence

The limit of a sequence is the limit of its general term.

For the sequence {an}:

If lim an = L    (L)
n → ∞  

then, the sequence { an } is convergent.

else

if the limit L does not exist, or infinite, the sequence is divergent.

An alternating sequence {(-1)nbn} with bn > 0 is:

• convergent to 0 , if lim [n → ∞] bn = 0
• divergent if lim [n → ∞] bn≠ 0 .



3. Properties of sequence


• lim [n → ∞] (an ± bn) = lim [n → ∞] an ± lim [n → ∞] bn

• lim [n → ∞] (c an) = c lim [n → ∞] an

• lim [n → ∞] (an x bn) = lim [n → ∞] an x lim [n → ∞] bn

• lim [n → ∞] (an / bn) = lim [n → ∞] an / lim [n → ∞] bn,
        provided lim [n → ∞] bn ≠ 0.

• lim [n → ∞] (an)p = (lim [n → ∞] an)p,
        provided an ≥ 0.



4. Related theorems of sequence


Theorem 0

If {an} is bounded and monotonic then {an} is convergent.

Note that if a sequence is not bounded and/or not monotonic does not say that it is divergent.

Theorem 1

Given the sequence {an} if we have a function f(x) such that f(n) = an and
lim [x → ∞] f(x) = L then lim [n → ∞] an = L

Squeeze Theorem

If an ≤ cn ≤ bn for all n ≥ N

for some N and lim [n → ∞] an = lim [n → ∞] bn = L then
lim [n → ∞] cn = L .

Theorem 2

If lim [n → ∞] |an| = 0 then
lim [n → ∞] an = 0.

Theorem 3

The sequence {rn} [ n = 0 → ∞]
does not have a limite when r ≥ 1,
has a limite 0 when - 1 ≤ r ≤ 1
has a limite 1 when r = 1
diverges for r ≥ 1

Theorem 4

For the sequence {an} if both lim [n → ∞] a2n = L and
lim [n → ∞] a2n+1 = L then {an} is convergent and
lim [n → ∞] an = L.








  


chimie labs
|
Physics and Measurements
|
Probability & Statistics
|
Combinatorics - Probability
|
Chimie
|
Optics
|
contact
|


© Scientificsentence 2010. All rights reserved.