Section 10.3 โ The Integral Test Positive Series: A series whose elements are non-negative. The Integral Test: Let f be a continuous, positive and decreasing function and ๐๐ = ๐(๐) for all positive integers, then โ โ ๐=๐ ๐๐ converges iff the improper integral ๐ ๐ ๐ ๐ ๐ converges If the improper integral diverges, then the series diverges. Section 10.3 โ The Integral Test Example: ๏ฅ 1 ๏ฅ n๏ฝ1 n ๏ซ 3 ๏ฒ 1 b lim ln| u|1 b๏ฎ ๏ฅ ๏ฅ 1 dx x๏ซ3 lim ๏ฒ b๏ฎ ๏ฅ lim ln| x ๏ซ 3|1 b b๏ฎ ๏ฅ b 1 1 dx x๏ซ3 u ๏ฝ x ๏ซ 3 du ๏ฝ dx lim ๏ln(b ๏ซ 3) ๏ญ ln(1 ๏ซ 3)๏ b๏ฎ ๏ฅ lim ๏ฒ b๏ฎ ๏ฅ ๏ฅ ๏ญ ln 4 b 1 1 du u ๏ divergent Section 10.3 โ The Integral Test Positive Series โ p-Series ๐=๐ ๐ ๐๐ p is a constant If p is ๏พ 1, then the series converges. If p is ๏ฃ 1, then the series diverges. โ ๐=๐ ๐ ๐๐ โ 1 1 ๐๐ฅ ๐ ๐ฅ ๐ lim ๐โโ 1 1 ๐๐ฅ ๐ ๐ฅ ๐ฅ โ๐+1 ๐ lim ๐โโ โ๐ + 1 1 1 1 lim ๐โ1 โ 1 1 โ ๐ ๐โโ ๐ 1 1 ๐ lim ๐โ1 1 โ ๐ ๐โโ ๐ฅ 1 Section 10.3 โ The Integral Test Positive Series Examples: โ ๐=๐ ๐ ๐๐ โ ๐ ๐=๐ ๐ โ๐ ๐ โ ๐๐๐๐๐๐ โ ๐ ๐ ๐=๐ ๐ ๐ ๐=๐>๐ ๐ โ ๐๐๐๐๐๐ converges ๐ ๐ = โค ๐ diverges ๐ Section 10.3 โ The Integral Test Harmonic Series โ ๐=๐ ๐ ๐ Positive Series ๐ โ ๐๐๐๐๐๐ ๐=๐โค๐ diverges Section 10.3 โ The Integral Test Section 10.4 โ Comparison Tests Positive Series Comparison Test (Ordinary Comparison Test) โ Let ๐ผ๐ be a positive series. ๐=๐ โ If ๐ฝ๐ is a positive convergent series and ๐ โค ๐ผ๐ โค ๐ฝ๐ ,then ๐=๐ the series โ If โ ๐=๐ ๐ผ๐ converges. ๐ฝ๐ is a positive divergent series and ๐ โค ๐ฝ๐ โค ๐ผ๐ ,then ๐=๐ the series โ ๐=๐ ๐ผ๐ diverges. Section 10.4 โ Comparison Tests Positive Series Examples: โ ๐=๐ ๐ ๐๐ + ๐ ๐ ๐ผ๐ = ๐ ๐ +๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ฝ๐ = ๐ = + โ + ๐ ๐ ๐ ๐ ๐ ๐ ๐๐ โค ๐๐ ๐ฝ๐ is a geometric series. ๐ ๐ ๐ = ,๐ = < ๐ ๐ ๐ ๐ฝ๐ is a convergent geometric series. โด ๐ผ๐ is a convergent series. ๐ +โฏ Section 10.4 โ Comparison Tests Limit Comparison Test Positive Series โ Let ๐ผ๐ be a positive series and ๐ผ๐ is a rational expression. ๐=๐ Choose ๐ฝ๐ to be a positive series. If ๐๐ lim ๐โโ ๐๐ = ๐ฟ, then โ ๐=๐ ๐ผ๐ will converge or diverge depending on โ ๐=๐ ๐ผ๐ will converge if โ ๐=๐ ๐ผ๐ will diverge if โ ๐=๐ ๐ฝ๐ โ ๐=๐ ๐ฝ๐ โ ๐=๐ ๐ฝ๐ provided ๐ < ๐ณ < โ. converges provided ๐ณ = ๐. diverges provided ๐ณ = โ. Section 10.4 โ Comparison Tests Positive Series Examples: โ ๐=๐ ๐ ๐๐ ๐ฝ๐ = ๐ +๐ ๐ ๐ ๐ lim ๐โโ ๐๐ +๐ ๐ ๐ ๐ 2 ๐ ๐ lim ๐โโ = ๐ ๐ ๐ ๐ ๐ ๐๐ ๐ ๐ ๐ +๐ ๐ ๐ฝ๐ is a p-series. ๐ ๐ ๐ lim ๐ ๐โโ ๐ ๐= <๐ โด ๐ ๐๐ ๐๐ 1 +๐ 3 ๐ ๐ โ ๐=๐ ๐ฝ๐ is div. ๐2 lim ๐โโ ๐2 + 2 1 3 ๐ ๐ lim ๐โโ ๐2 + 2 1 3 =1 ๐ณ=๐ 0 < 1 < โ, โ ๐=๐ ๐ฝ๐ is div. โด โ ๐=๐ ๐ผ๐ is div.
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