Calculus II - Sequences

19 flashcards · Mathematics · by Harrison Holbrook

Sequence
A sequence is a function whose domain is the set of positive integers and whose range is a subset of the real numbers
sns_nsn
In sns_nsn notation, the sequence (or series) is defined through a formula for the nth term of the sequence
Alternating sequence
A sequence where each subsequent term changes sign, commonly in the form (−1)nan(-1)^na^n(−1)nan
Convergent sequence
Suppose LLL is a real number and sns_nsn is a sequence. sns_nsn is said to converge to the real number LLL if, for any number 0"> ϵ>0\epsilon > 0ϵ>0 , there is a positive integer NNN so that ∣sn−L∣<ϵ|s_n-L|<\epsilon∣sn−L∣<ϵ for all integers N"> n>Nn>Nn>N . If sns_nsn converges to LLL , then LLL is called the limit of the sequence and we write lim⁡n→∞sn=L\lim_{n\to\infty}s_n=Llimn→∞sn=L . If a sequence converges, it is said to be convergent, otherwise it is said to be divergent. Therefore,…
Properties of convergent sequences
The properties of limits applies to all convergent sequences (ex. constant multiple rule, sum rule, quotient rule)
Limit of a function of a sequence
Suppose sns_nsn is a sequence of real numbers. If lim⁡n→∞sn=L\lim_{n\to\infty}s_n=Llimn→∞sn=L and fff is a function that is continuous at LLL and defined for all numbers sns_nsn , then lim⁡n→∞f(sn)=f(L)\lim_{n\to\infty}f(s_n)=f(L)limn→∞f(sn)=f(L)
Related function to a sequence
A related function fff of the sequence sns_nsn has the following two properties fff is defined on the open interval (0,∞)(0,\infty)(0,∞) - its domain is the set of all positive real numbers f(n)=snf(n)=s_nf(n)=sn for all integers n≥1n \geq 1n≥1
Limit of a related function
Suppose sns_nsn is a sequence of real numbers, fff is a related function of sns_nsn , and LLL is a real number. If lim⁡x→∞f(x)=L\lim_{x\to\infty} f(x)=Llimx→∞f(x)=L , then lim⁡n→∞sn=L\lim_{n\to\infty} s_n=Llimn→∞sn=L
The squeeze theorem for sequences
Suppose ana_nan , bnb_nbn , and sns_nsn are sequences and NNN is a positive integer. If an≤sn≤bna_n \leq s_n \leq b_nan≤sn≤bn for every integer N"> n>Nn>Nn>N , and if lim⁡n→∞an=lim⁡n→∞bn=L\lim_{n\to\infty} a_n=\lim_{n\to\infty} b_n=Llimn→∞an=limn→∞bn=L , then lim⁡n→∞sn=L\lim_{n\to\infty} s_n=Llimn→∞sn=L
Sequence divergence
A sequence sns_nsn diverges to infinity if lim⁡n→∞sn=∞\lim_{n\to\infty} s_n=\inftylimn→∞sn=∞
Convergence or divergence of geometric sequence
The geometric sequence sn=rns_n=r^nsn=rn , where rrr is a real number Converges to 000 if −1<r<1-1<r<1−1<r<1 Converges to 111 if r=1r=1r=1 Diverges for all other numbers ( r<−1r<-1r<−1 , 1"> r>1r>1r>1 )
Sequence bounded from above
A sequence sns_nsn is said to be bounded from above if every term of the sequence is less than some real number MMM , that is sn<Ms_n < Msn<M for all nnn .
Sequence bounded from below
A sequence sns_nsn is said to be bounded from below if every term of the sequence is greater than some real number MMM , that is M"> sn>Ms_n >Msn>M for all nnn .
Bounded sequence
A sequence sns_nsn is bounded if it is both bounded from above and below. For a bounded sequence, there is a positive number KKK for which ∣sn∣<K|s_n| < K∣sn∣<K for all n≥1n \geq 1n≥1
Boundedness theorem
A convergent sequence is bounded (notably, not all bounded sequences converge, implying the converse is not always true)
Bounded test for divergence of a sequence
If a sequence is not bounded from above or if it is not bounded from below, then the sequence diverges
Monotonic sequences
A sequence sns_nsn is said to be Increasing if sn<sn+1s_n < s_{n+1}sn<sn+1 for all n≥1n \geq 1n≥1 Nondecreasing if sn≤sn+1s_n \leq s_{n+1}sn≤sn+1 for all n≥1n \geq 1n≥1 Decreasing if s_{n+1}"> sn>sn+1s_n > s_{n+1}sn>sn+1 for all n≥1n \geq 1n≥1 Nonincreasing if sn≥sn+1s_n \geq s_{n+1}sn≥sn+1 for all n≥1n \geq 1n≥1 If any of these conditions hold, the sequence is said to be monotonic
Methods to determine a monotonic sequence
Use either algebra, a ratio of sns_nsn to sn+1s_{n+1}sn+1 , or the derivative of a related function of sns_nsn
Convergence of a monotonic sequence
A bounded monotonic sequence converges