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Mat136 Series Tests
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Exam of 2 pages for the course MATH at MATH (Mat136 Series Tests)
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Mat136 Series TestsOnly the latter terms of series influence convergence.
lim(n→∞){sₙ}=L, finite number and converge
If the limit is +/-∞ or DNE - ANS-Convergence of Infinite Sequences
For +/- series and gives the sum (L value)
Take lim(n→∞) {Sₙ}= L, if finite converges, otherwise diverges.
{Sₙ} is a sequence of partial sums
{Sₙ}=a₁+a₂+a₃...aₙ of series ∑aₙ - ANS-Sequence of Partial Sums
For +/- series and gives the sum
Given series: ∑arⁿ=ar⁰+ar¹+ar²...
; |r|≥1→diverges, |r|<1→converges
Sum = a/(1-r), if series converges - ANS-Geometric Series
For + Series
Given series ∑1/(n^p), where n∈I, p∈R, p>0
Series diverges for p≤1 and converges for p>1 - ANS-P-Series
For + , continuous, decreasing series
If Improper Integral diverges, Infinite series diverges or converge → converge
Integral to infinity:∫aₙ for series∑aₙ - ANS-Integral Test
For +/- series
Given lim(n→∞) of series aₙ≠0→diverges
If limit = 0, then no information - ANS-Test of Divergence
For + Series
Assume bₙ≥aₙ;
If ∑bₙ converges→∑aₙ converges
iIf ∑aₙ diverges→∑bₙ diverges - ANS-The "Comparison" Test
For+ series
If lim(n→∞) aₙ/bₙ=C (C>0); Both converge or diverge
If ±∞,0, or DNE, then no information obtained
*Useful for polynomial/polynomial or polynomial like fractions.
*Choose highest order polynomials for Bₙ - ANS-Limit Comparison Test
For +/- Series
Given ∑(-1)ⁿbₙ = -b₁+b₂-b₃...
Converges if bₙ₊₁≤bₙ and lim(n→∞) bₙ=0
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