Calc Formula Sheet - Divergence test If the limnan0 then the series an diverges If limnan=0 then the divergence test fails Geometric Series Test The

Calc Formula Sheet - Divergence test If the limnan0 then...

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Convergence of Geometric SeriesTo determine what each geometric series converges to use the following equations: If |r| < 1, then the series converges:= ∞n0crn= -11 ror=nmcrn= -crm1 rIf |r| > 1, then the series diverges but the partial sum is:== ( -+ ) -nmNcrnc 1 rN 1 1 rIntegral TestFor the series, = ∞n0an, an0for all n then define a function, f(x), such that f(n)=anand if mfxdxconverges then the series anconverges as wellP-TestThe sum = ∞n11npconverges if p > 1 and diverges if p ≤ 1Comparison Test 0 ≤ an≤bnIf = ∞n0bnconverges then = ∞n0analso converges If = ∞n0andiverges then = ∞n0bnalso diverges. Limit Comparison TestThere is a series = ∞n0an. Find the simplified version of an, call that series bn = ∞nbnconverges then = ∞nanconverges2.If L > 0 then = ∞naniff= ∞nbnconvergesAbsolute Convergenceanis called absolutely convergent if anconvergesConditional Convergence anis called conditionally convergent if anconverges and andivergesLeibniz Alternating Series TestIff : i)an > an+1 and ii)→∞=limnan(an converges to 0)Then the following alternating series is said to conditionally converge:= ∞nQan, where Qis (-1)n, (-1)n+1, (-1)n-1, etc. Root Test
=→∞Llimnnani)If L< 1 then anconverges absolutely ii) If L> 1 then andiverges iii) If L= 1 then the test fails and is inconclusiveRatio Test=→∞+ρlimnan 1aniv) If ρ< 1 then anconverges absolutely v) If ρ> 1 then andiverges vi) If ρ= 1 then the test fails and is inconclusive