Chapter 4: Seriessha Uni. of Sci & Tech)FCV&ITAugust19,20193/50MinsLlChar
Chapter 4: Series Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 3 / 50
s4.1 Series of complex numbers and series of complexfunctionsaUni.ofSci&Tech)FCV&ITAugust19,20194/50
§4.1 Series of complex numbers and series of complex functions Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 4 / 50
SeoennumberSeqencesofcomplexnumbersAsequenceQnofcomplexnumbersQ1=a1+ibi,Q2=a2+ib2,.,Qn=an+ibn，isdenoted by [an]Definition (Convergence)It is said to converge to a complex number , if for all e> O, there is anintegerN>O,suchthatn≥N impliesthatan-a<e.Convergenceofantoαisdenotedbyan→Q.Definition(Cauchy sequence)A sequence Qn converges if and only if it is a Cauchy sequence:for every>0,thereisanN,suchthatn,m>N impliesthatan-Qm/<.limn-→oQn=α=a+ib iffliman=aandlimbn=b.n-→0n→xFCV&ITAugust 19,20195/50tha Uni.of Sci&Tech)MineLilChan
Seqences of complex numbers Seqences of complex numbers A sequence αn of complex numbers: α1 = a1 + ib1, α2 = a2 + ib2, · · · , αn = an + ibn, · · · is denoted by {αn} Definition (Convergence) It is said to converge to a complex number α, if for all ε > 0, there is an integer N > 0, such that n ≥ N implies that |αn − α| < ε. Convergence of αn to α is denoted by αn → α. Definition (Cauchy sequence) A sequence αn converges if and only if it is a Cauchy sequence: for every ε > 0, there is an N, such that n, m ≥ N implies that |αn − αm| < ε. limn→∞ αn = α = a + ib iff limn→∞ an = a and limn→∞ bn = b. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 5 / 50
numbersSeoerSeriesof complexnumbersAn infinite seriesisa1+a2++an+.*Sn =h=1 Qk is called the partial sums, it is a sequence.Definition (Convergence)the series k-, Qk is said to converges to S and we write k=, Qk = S,ifthesequenceofpartialsumsSnconvergestoSTheorem (Carchycriterion)k=i ak converges iff for every e>O, there is an N such that n ≥Nan+pimpliesthat<eforallp=1.2,3....一k=n+10kTheoremk=1 ak converges iff both k=1 ak and k=, bk convergeFCV&ITAugust 19,2019sha Uni. of Sci&Tech)6/50MingLi(Chan)
Seqences of complex numbers Series of complex numbers An infinite series is α1 + α2 + · · · + αn + · · · Sn = Pn k=1 αk is called the partial sums, it is a sequence. Definition (Convergence) the series P∞ k=1 αk is said to converges to S and we write P∞ k=1 αk = S, if the sequence of partial sums Sn converges to S. Theorem (Carchy criterion) P∞ k=1 αk converges iff for every ε > 0, there is an N such that n ≥ N implies that Pn+p k=n+1 αk < ε for all p = 1, 2, 3, · · · Theorem P∞ k=1 αk converges iff both P∞ k=1 ak and P∞ k=1 bk converge. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 6 / 50
numbersExample(4.1.1)Determine whether the series k=1 (+)convergesSolution.No,because=divergesTheorem (4.1.2)If =1 ak converges, then ak → 0.Proof.Z=1 k converges, iff for every e >0, there is an N, such that n ≥ Nmple that +, fo all =1,3..Asa particularcaseof Cauchycriterionwithp=1口FCV&ITha Uni.of Sci &Tech)August19,20197/50MineLilCh
Seqences of complex numbers Example (4.1.1) Determine whether the series P∞ k=1 1 k + i 2 k converges. Solution. No, because P∞ k=1 1 k diverges. Theorem (4.1.2) If P∞ k=1 αk converges, then αk → 0. Proof. P∞ k=1 αk converges, iff for every ε > 0, there is an N, such that n ≥ N implies that Pn+p k=n+1 < ε, for all p = 1, 2, 3, · · · . As a particular case of Cauchy criterion with p = 1. Ming Li (Changsha Uni. of Sci & Tech) FCV & IT August 19, 2019 7 / 50