Tiêu đề Magic Table For Testing Convergence or Divergence of Infinite Series.pdf ✅

SANJAY GUPTA Limit Comprison Test Let ∑ ࢔ࢇ and ∑ ࢔࢈ are series with positive terms such that ஶ→࢔ܕܑܔ ࢔ࢇ ࢔࢈ ࢒ = (i) If ࢒ < 0, then both the series converge or diverge together (ii) If ࢒ ૙ = and ∑ ࢔࢈ converges, then ∑ ࢔ࢇ converges (iii) If ࢒ = + ∞and ∑ ࢔࢈ diverges, then ∑ ࢔ࢇ diverges This test is easier to apply than the Direct Comparison Test, but requires some skill in choosing the series ∑ ࢔࢈ for comparison. e.g. ∑ ඥ࢔૜ା૚ ૛ା૛࢔૝ା૜࢔૜ (say (࢔ࢇ∑ = = ࢔࢈ Choose ૜࢔ඥ = ૜࢔૜ ૚ ࢔૜ ૜ ૛ This Test is applied only when ࢔ࢇ does not involves any power of x involving n and when ࢔ࢇ does not involve factorials. D'Almbert Ratio Test Let ∑ ࢔ࢇ be a series with positive terms such that ܔܑ࢔ܕ→ஶ ૚శ࢔ࢇ ࢔ࢇ .࢒ = (i) If ࢒ > 1 , then ∑ ࢔ࢇ converges (ii) If ࢒ < 1 , then ∑ ࢔ࢇ diverges (iii) If ࢒ ૚ = , then another test must be used Try this test when ࢔ࢇ involves factorials or n th powers. If Ratio Test fails i.e. ࢒૚ = , then (i) use Raabe’s or Gauss’s Test if ૚శ࢔ࢇ ࢔ࢇ does not involve the .ࢋ number (ii) use Logrithmic Test if ૚శ࢔ࢇ ࢔ࢇ involves the number ࢋ. Raabe’s Test Let ∑ ࢔ࢇ be a series with positive terms such that ቀ࢔ஶ→࢔ܕܑܔ ࢔ࢇ ૚శ࢔ࢇ ࢒ = ቁ ૚ − (i) If ࢒ < 1 , then ∑ ࢔ࢇ converges (ii) If ࢒ > 1 , then ∑ ࢔ࢇ diverges Raabe’s Test is used when Ratio Test fails and the ratio ࢔ࢇ ૚శ࢔ࢇ does not involve the number ࢋ. Gauss’s Test Let ∑ ࢔ࢇ be a series with positive terms such that ࢔ࢇ ૚శ࢔ࢇ = ૚ + ࣆ ࢔ + ભ ቀ ૚ ૛࢔ ቁ where ભ ቀ ૚ ૛࢔ ቁ stands for the terms of order ૚ ૛࢔ and higher power of ૚ ࢔ (i) If ࣆ < 1 , then ∑ ࢔ࢇ converges (ii) If ࣆ ૚ ≥ , then ∑ ࢔ࢇ diverges Gauss’s Test never fails as the series diverges for ࣆ૚ = . Moreover the test is applied after the failure of Ratio Test and when it is possible to expand ࢔ࢇ ૚శ࢔ࢇ in powers of ૚ ࢔ by usingBinomial Theorem or by any other methods. Gauss’s Test is used when Ratio Test fails and the ratio ࢔ࢇ ૚శ࢔ࢇ does not involve the number ࢋ. Logrithmic Test Let ∑ ࢔ࢇ be a series with positive terms such that ቀࢍ࢕࢒࢔ஶ→࢔ܕܑܔ ࢔ࢇ ૚శ࢔ࢇ ࢒ = ቁ (i) If ࢒ < 1 , then ∑ ࢔ࢇ converges (ii) If ࢒ > 1 , then ∑ ࢔ࢇ diverges (iii) If ࢒ ૚ = , then the test fails Logrithmic Test is used when Ratio Test fails and the ratio ࢔ࢇ ૚శ࢔ࢇ involves the number ࢋ. e.g. ∑ ! ࢔ ࢞ ࢔(૚ା࢔) ࢔ 0 < ࢞ , devsamajcollege.blogspot.in SANJAY GUPTA, DEV SAMAJ COLLEGE FOR WOMEN, FEROZEPUR CITY
SANJAY GUPTA Cauchy's Root Test Let ∑ ࢔ࢇ be a series with positive terms such that ܔܑ࢔ܕ→ஶ(࢔ࢇ ) ૚ ࢒ = ࢔ (i) If ࢒ > 1 , then ∑ ࢔ࢇ converges (ii) If ࢒ < 1 , then ∑ ࢔ࢇ diverges (iii) If ࢒ ૚ = , then another test must be used This test is the most accurate, but not the easiest to use in many situation. Try this test when ࢔ࢇ involves n th powers e.g. ∑ ቀ ૚ା࢔ ૚ା࢔૛ ቁ ࢔ Alternating Series Test ( Leibnitz’s Theorem) The series ∑(−૚) ⋯ + ૝ࢇ − ૜ࢇ + ૛ࢇ − ૚ࢇ = ࢔ࢇ૚ି࢔ or ∑(−૚) if converges ⋯ −૝ ࢇ + ૜ࢇ − ૛ࢇ + ૚ࢇ− = ࢔ࢇ࢔ ૙ = ࢔ࢇ ஶ→࢔ܕܑܔ (ii (݊ ∀ 0 > ࢔ࢇ (i( (iii) The sequence {࢔ࢇ {is monotonically decreasing i.e. ࢔ࢇ ≤ ࢔ࢇା૚ ∀ ࢔ ∋ ࡺ This test applies only to alternating series. The alternating series diverges (oscillates finitely) ૙ =/ ࢔ࢇ ஶ→࢔ܕܑܔ if e.g. The alternating series ∑(−૚) ૜ ࢔ ૚ା࢔૝ converges but ∑(−૚) ૛ି࢔૜ ࢔ ૞ା࢔૝ diverges. Absolute Convergence and Conditional Convergence Let ∑ ࢔ࢇ be a series with non - zero terms. (i) If ∑|࢔ࢇ |converges , then ∑ ࢔ࢇ converges absolutely (ii) If ∑ ࢔ࢇ converges but ∑|࢔ࢇ |diverges , then ∑ ࢔ࢇ converges Conditionally Note that every absolutely convergent series is convergent. e.g. ∑(−૚) ૚ି࢔ ૚ ! (૚ା࢔૛) is absolutely convergent and ∑(−૚) ૚ି࢔ ૚ (૚ା࢔૛) is conditionally convergent Cauchy's Condensation Test If ࢌ)࢔ (is a positive monotonically decreasing function of n i.e. (࢔ ≤ (ࢌ)࢔ + ∀ ૙ ≤ (૚ ࢔ , then the series ∑ ࢌ)࢔ (and ∑ ૛ ૛)ࢌ ࢔ ࢔ (converges or diverges together. e.g. ∑ ૚ (࢔ࢍ࢕࢒)࢔ ࢖ ஶ ૛ୀ࢔ converges if ࢖ < 1 and diverges if ࢖૚ ≥ READ MMS RADIANT FOR SURE SUCCESS IN MAY 2018 EXAMS devsamajcollege.blogspot.in SANJAY GUPTA, DEV SAMAJ COLLEGE FOR WOMEN, FEROZEPUR CITY