Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

5.1Sequences

• To determine the convergence of a sequence given by an explicit formula $an=f(n),an=f(n),$ we use the properties of limits for functions.
• If ${an}{an}$ and ${bn}{bn}$ are convergent sequences that converge to $AA$ and $B,B,$ respectively, and $cc$ is any real number, then the sequence ${can}{can}$ converges to $c·A,c·A,$ the sequences ${an±bn}{an±bn}$ converge to $A±B,A±B,$ the sequence ${an·bn}{an·bn}$ converges to $A·B,A·B,$ and the sequence ${an/bn}{an/bn}$ converges to $A/B,A/B,$ provided $B≠0.B≠0.$
• If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.
• If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.
• The geometric sequence ${rn}{rn}$ converges if and only if $|r|<1|r|<1$ or $r=1.r=1.$

5.2Infinite Series

• Given the infinite series
$∑n=1∞an=a1+a2+a3+⋯∑n=1∞an=a1+a2+a3+⋯$

and the corresponding sequence of partial sums ${Sk}{Sk}$ where
$Sk=∑n=1kan=a1+a2+a3+⋯+ak,Sk=∑n=1kan=a1+a2+a3+⋯+ak,$

the series converges if and only if the sequence ${Sk}{Sk}$ converges.
• The geometric series $∑n=1∞arn−1∑n=1∞arn−1$ converges if $|r|<1|r|<1$ and diverges if $|r|≥1.|r|≥1.$ For $|r|<1,|r|<1,$
$∑n=1∞arn−1=a1−r.∑n=1∞arn−1=a1−r.$
• The harmonic series
$∑n=1∞1n=1+12+13+⋯∑n=1∞1n=1+12+13+⋯$

diverges.
• A series of the form $∑n=1∞[bn−bn+1]=[b1−b2]+[b2−b3]+[b3−b4]+⋯+[bn−bn+1]+⋯∑n=1∞[bn−bn+1]=[b1−b2]+[b2−b3]+[b3−b4]+⋯+[bn−bn+1]+⋯$
is a telescoping series. The $kthkth$ partial sum of this series is given by $Sk=b1−bk+1.Sk=b1−bk+1.$ The series will converge if and only if $limk→∞bk+1limk→∞bk+1$ exists. In that case,
$∑n=1∞[bn−bn+1]=b1−limk→∞(bk+1).∑n=1∞[bn−bn+1]=b1−limk→∞(bk+1).$

5.3The Divergence and Integral Tests

• If $limn→∞an≠0,limn→∞an≠0,$ then the series $∑n=1∞an∑n=1∞an$ diverges.
• If $limn→∞an=0,limn→∞an=0,$ the series $∑n=1∞an∑n=1∞an$ may converge or diverge.
• If $∑n=1∞an∑n=1∞an$ is a series with positive terms $anan$ and $ff$ is a continuous, decreasing function such that $f(n)=anf(n)=an$ for all positive integers $n,n,$ then
$∑n=1∞anand∫1∞f(x)dx∑n=1∞anand∫1∞f(x)dx$

either both converge or both diverge. Furthermore, if $∑n=1∞an∑n=1∞an$ converges, then the $NthNth$ partial sum approximation $SNSN$ is accurate up to an error $RNRN$ where $∫N+1∞f(x)dx
• The p-series $∑n=1∞1/np∑n=1∞1/np$ converges if $p>1p>1$ and diverges if $p≤1.p≤1.$

5.4Comparison Tests

• The comparison tests are used to determine convergence or divergence of series with positive terms.
• When using the comparison tests, a series $∑n=1∞an∑n=1∞an$ is often compared to a geometric or p-series.

5.5Alternating Series

• For an alternating series $∑n=1∞(−1)n+1bn,∑n=1∞(−1)n+1bn,$ if $bk+1≤bkbk+1≤bk$ for all $kk$ and $bk→0bk→0$ as $k→∞,k→∞,$ the alternating series converges.
• If $∑n=1∞|an|∑n=1∞|an|$ converges, then $∑n=1∞an∑n=1∞an$ converges.

5.6Ratio and Root Tests

• For the ratio test, we consider
$ρ=limn→∞|an+1an|.ρ=limn→∞|an+1an|.$

If $ρ<1,ρ<1,$ the series $∑n=1∞an∑n=1∞an$ converges absolutely. If $ρ>1,ρ>1,$ the series diverges. If $ρ=1,ρ=1,$ the test does not provide any information. This test is useful for series whose terms involve factorials.
• For the root test, we consider
$ρ=limn→∞|an|n.ρ=limn→∞|an|n.$

If $ρ<1,ρ<1,$ the series $∑n=1∞an∑n=1∞an$ converges absolutely. If $ρ>1,ρ>1,$ the series diverges. If $ρ=1,ρ=1,$ the test does not provide any information. The root test is useful for series whose terms involve powers.
• For a series that is similar to a geometric series or $p−series,p−series,$ consider one of the comparison tests.
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