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Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

Key Concepts

5.1 Sequences

  • 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 B0.B0.
  • 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.

5.2 Infinite Series

  • Given the infinite series
    n=1an=a1+a2+a3+n=1an=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=1arn1n=1arn1 converges if |r|<1|r|<1 and diverges if |r|1.|r|1. For |r|<1,|r|<1,
    n=1arn1=a1r.n=1arn1=a1r.
  • The harmonic series
    n=11n=1+12+13+n=11n=1+12+13+

    diverges.
  • A series of the form n=1[bnbn+1]=[b1b2]+[b2b3]+[b3b4]++[bnbn+1]+n=1[bnbn+1]=[b1b2]+[b2b3]+[b3b4]++[bnbn+1]+
    is a telescoping series. The kthkth partial sum of this series is given by Sk=b1bk+1.Sk=b1bk+1. The series will converge if and only if limkbk+1limkbk+1 exists. In that case,
    n=1[bnbn+1]=b1limk(bk+1).n=1[bnbn+1]=b1limk(bk+1).

5.3 The Divergence and Integral Tests

  • If limnan0,limnan0, then the series n=1ann=1an diverges.
  • If limnan=0,limnan=0, the series n=1ann=1an may converge or diverge.
  • If n=1ann=1an 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=1anand1f(x)dxn=1anand1f(x)dx

    either both converge or both diverge. Furthermore, if n=1ann=1an converges, then the NthNth partial sum approximation SNSN is accurate up to an error RNRN where N+1f(x)dx<RN<Nf(x)dx.N+1f(x)dx<RN<Nf(x)dx.
  • The p-series n=11/npn=11/np converges if p>1p>1 and diverges if p1.p1.

5.4 Comparison Tests

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

5.5 Alternating Series

  • For an alternating series n=1(−1)n+1bn,n=1(−1)n+1bn, if bk+1bkbk+1bk for all kk and bk0bk0 as k,k, the alternating series converges.
  • If n=1|an|n=1|an| converges, then n=1ann=1an converges.

5.6 Ratio 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=1ann=1an 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=1ann=1an 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 pseries,pseries, consider one of the comparison tests.
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