Convergence Test Chart

# Convergence Test Chart - Ratio a n n=1 I a n 1 I...

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Test Series Condition(s) of  Convergence Condition(s) of  Divergence Comment n th-Term Σ a n n=1 lim a n 0 n →∞ This test cannot be  used to show  convergence Geometric Series Σ ar n n=0 I   r I <   1 I   r I   ≥ 1                        a Sum:   S = ----- 1 - r Telescoping  Series Σ (b n - b n+ 1 ) n=1 lim b n = L n →∞ Sum:  S = b 1 - L p -Series 1 Σ --- n=1 n p P > 1 P ≤ 1 Alternating  Series Σ (- 1 ) n-1 a n n=1 0 < a n+1 a n and lim a n = 0 n →∞ Remainder:  I   R N    a N+1 Integral  f  is  continuous, positive,  and decreasing) Σ a n , n=1 a n = f (n) ≥ 0 f(x) dx converges 1 f(x) dx diverges 1 Remainder:                    0 < R N < f(x) dx N Root Σ a n n=1 _____ lim n  I   a n I    < 1 n →∞ _____ lim n  I   a n I    > 1 n →∞ Test is inconclusive if  _____ lim n  I   a n I    = 1 n →∞ Ratio
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Unformatted text preview: Ratio a n n=1 I a n+ 1 I lim------- &amp;lt; 1 n I a n I I a n+ 1 I lim------- &amp;gt; 1 n I a n I Test is inconclusive if I a n+ 1 I lim------- = 1 n I a n I Direct Comparison ( a n , b n &amp;gt; 0 ) a n n=1 0 &amp;lt; a n b n and b n converges n=1 0 &amp;lt; b n a n and b n diverges n=1 Limit Comparison ( a n , b n &amp;gt; ) a n n=1 a n lim------- = L &amp;gt; 0 n b n and b n converges n=1 a n lim----- = L &amp;gt; 0 n b n and b n diverges n=1...
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