Series&Test&&Convergence& &Divergence&1-Divergence&Test&lim$โ&๐$โ 0!Only!a!test!for!divergence!2-Geometric&Series&|๐|<1!|๐|โฅ1!/๐๐$0=0/๐๐$23&$430=0๐10โ0๐0๐๐0|๐|<1&$48!3-Integral&Test&9๐(๐ฅ)๐๐ฅ0๐๐๐๐ฃ๐๐๐๐๐ &3!9๐(๐ฅ)๐๐ฅ0๐๐๐ฃ๐๐๐๐๐ &3!4-p-series&Test&๐ > 1!๐ โค 1!series!has!form!โ3$N&$43!5-Direct&Comparison&Test&๐$0โค0๐$!and!โ๐$converges!๐$0โฅ0๐$!and!!โ๐$!diverges!original!series!=!โ ๐$! ;!Comparison!series!=!โ ๐$!6-Limit&Comparison&Test&lim$โ&๐$๐$0=0๐ฟ0โ 000&0/๐$00๐๐๐๐ฃ๐๐๐๐๐ !OR!lim$โ&๐$๐$=00&0/๐$00๐๐๐๐ฃ๐๐๐๐๐ !lim$โ&๐$๐$0=0๐ฟ0โ 000&0/๐$00๐๐๐ฃ๐๐๐๐๐ !OR!limโ&๐$๐$= โ0 orthe limit soesn:t &/๐$ 0๐๐๐ฃ๐๐๐๐๐ !your series!=!โ ๐$! ;!!Comparison!series!=!โ ๐$!7-Alternating&Series&Test&lim$โ&๐$=0!and!!๐$L3โค๐$00โ๐!โ๐$=โ(โ1)$๐$!;!If!series!is!convergent,!โ(โ1)$๐$=0๐ 0 =0๐ $0+0๐ $!!and!|๐ $|โค๐$L3!8-Absolute&Convergence&/|๐$|&$480๐๐๐๐ฃ๐๐๐๐๐ !9-Ratio&Test&lim$โ&T๐$L3๐$T=๐ฟ < 1!Then!series!is!absolutely!convergent.!lim$โ&T๐$L3๐$T=๐ฟ > 1!lim$โ&Tthe test is not useful if!๐$L3๐$T=๐ฟ = 1!10-Root&Test&lim$โ&U|๐$|V=๐ฟ < 1!Then!series!is!absolutely!convergent.!lim$โ&U|๐$|V=๐ฟ > 1! the test is not useful if lim$โ&U|๐$|V=๐ฟ = 1!Interval&of&Convergence&for&a&Power&Series&โ๐๐&๐4๐=0โ๐๐(๐โ๐)๐&๐4๐&Use!Ratio!Test!or!Root!Test!and!force!convergence.!โข If! lim$โ&\]V^_]V\=00,๐กโ๐๐00 < 10โ๐ฅ!!Then!๐ =โ00๐๐๐0๐ผ๐๐ก๐๐๐ฃ๐๐0๐๐0๐ถ๐๐๐ฃ๐๐๐๐๐๐๐0 =0(โโ,โ)!โข If!lim$โ&\]V^_]V\=0โ, ๐กโ๐๐0โโฎ10๐๐๐0๐๐๐ฆ00๐ฅ.!Then!๐ = 00๐๐๐0๐ผ๐๐ก๐๐๐ฃ๐๐0๐๐0๐ถ๐๐๐ฃ๐๐๐๐๐๐๐0๐๐ 0๐๐ข๐ ๐ก0๐ฅ0 =0๐.!โขlim$โ&\]V^_]V\=0๐|๐ฅโ๐|, ๐กโ๐๐0๐๐๐๐๐0๐๐๐๐ฃ๐๐๐๐๐๐๐0๐๐๐0๐ ๐๐๐ฃ๐0๐๐0๐๐ ๐๐๐๐ก๐0|๐ฅโ๐|0๐ก๐0๐๐๐ก0๐ .!olim$โ&\]V^_]V\=0๐|๐ฅโ๐|<1,000|๐ฅโ๐|<3k0=0๐ o Initial!Interval!of!Conv e r ge n c e !is!|๐ฅโ๐|<๐ , ๐๐0โ๐ < ๐ฅโ๐ <๐ , ๐คโ๐๐โ0๐๐๐๐ข๐๐๐ 0๐ก๐0๐โ๐ < ๐ฅ < ๐+๐ o Use!series!tests!to!test!the!end!points!of!this!interval!for!convergence.!Include!endpoin ts!th a t!co n v e r g e ,!an d !exclude!the!endpoints!that!diverge!in!the!Final!Interval!of!Convergence. is continuous , positive and decreasing on any value greater than or equal 1.with positive terms with positive termswith positive termspositive termsA series โ ๐๐ is called conditionally convergent ifโ ๐๐ converges but โ|๐๐| diverges.Differentiation&and&Integration&of&Power&Series&If!the!power!ser ie s!โ๐$(๐ฅโ๐)$0has!a!radius!of!convergence!๐ > 0,!then!the!function!๐(๐ฅ)!defined!by!๐(๐ฅ)0=0๐8+0๐3(๐ฅโ๐)+๐m(๐ฅโ๐)m+...= /๐$(๐ฅโ๐)$&$48!is!differentiable!on!the!interva l!(๐โ๐ ,๐+๐ )!and!(1) ๐n(๐ฅ)0=0โ๐๐$(๐ฅโ๐)$23&$43(2)โซ๐(๐ฅ)๐๐ฅ = ๐ถ+โ๐$(o2])V^_$L3&$48The!radii!of!convergence!of!the!power!series!in!(1)!and!(2)!are!both!R.!!Maclaurin&Series&Radius&of&Convergence&11โ๐ฅ=/๐ฅ$&$480=010+0๐ฅ0+0๐ฅm+0๐ฅp+...!๐ =1!๐o=0/๐ฅ$๐!&$480=01+๐ฅ1!+๐ฅm2!+๐ฅp3!+...!๐ =โ!๐ ๐๐๐ฅ0 =0/(โ1)$๐ฅm$L3(2๐+1)!&$48=0๐ฅโ๐ฅp3!+๐ฅt5!โ๐ฅv7!+...!๐ =โ!๐๐๐ ๐ฅ0 =0/(โ1)$๐ฅm$(2๐)!&$48=01โ๐ฅm2!+๐ฅx4!โ๐ฅz6!+...!๐ =โ!๐ก๐๐23๐ฅ0 =0/(โ1)$๐ฅm$L3(2๐+1)&$48=0๐ฅโ๐ฅp3+๐ฅt5โ๐ฅv7+...!๐ =1!๐๐(1+๐ฅ) =0/(โ1)$23๐ฅ$๐&$43=0๐ฅโ๐ฅm2+๐ฅp3โ๐ฅx4+...!๐ =1!(1+๐ฅ)|=0โ}๐๐โข๐ฅ$=01+๐๐ฅ+|(|23)m!&$48๐ฅm+|(|23)(|2m)p!๐ฅp+...!๐ =1!If!๐(๐ฅ)!has!a!power!series!representation!(expansion)!at!๐,!that!is,!if!๐(๐ฅ)0=0/๐$(๐ฅโ๐)$&$4800000000|๐ฅโ๐|<๐ !then!its!coefficients!a re !g iv e n !by !th e !for mula!๐$=0๐($)(๐)๐!The!nth!degree!Taylor!polynomial!of!๐(๐ฅ)0๐๐ก0๐!(or!order!n!centered!at!a):!๐$(๐ฅ)0=0๐8+๐3(๐ฅโ๐)0+๐m(๐ฅโ๐)m+...+๐$(๐ฅโ๐)$!!where!0๐$=0๐($)(๐)๐!Taylor's!Inequality!If!โข๐($L3)(๐ฅ)โขโค๐0๐๐๐0|๐ฅโ๐|โค๐!!then!the!remainder!๐ $(๐ฅ)!of!the!Taylor!series!satisfies!the!inequality!|๐ $(๐ฅ)|โคฦ($L3)!|๐ฅโ๐|$L3!!for!|๐ฅโ๐|โค๐!๐(๐ฅ)0=0๐$(๐ฅ)+๐ $(๐ฅ)!!!If! lim$โ&๐ $(๐ฅ) 0 = 000๐๐๐0|๐ฅ โ ๐| < ๐ !then!f(x)!is!equal!to!the!sum!of!its Taylor!series.!on!|๐ฅ โ ๐| <
View Full Document
Unlocking...