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Key Equations

Key Equations

Harmonic series ∑n=1∞1n=1+12+13+14+⋯∑n=1∞1n=1+12+13+14+⋯
Sum of a geometric series ∑n=1∞arn−1=a1−rfor|r|<1∑n=1∞arn−1=a1−rfor|r|<1
Divergence test Ifan↛0asn→∞,∑n=1∞andiverges.Ifan↛0asn→∞,∑n=1∞andiverges.
p-series ∑n=1∞1np{converges ifp>1diverges ifp≤1∑n=1∞1np{converges ifp>1diverges ifp≤1
Remainder estimate from the integral test ∫N+1∞f(x)dx<RN<∫N∞f(x)dx∫N+1∞f(x)dx<RN<∫N∞f(x)dx
Alternating series ∑n=1∞(−1)n+1bn=b1−b2+b3−b4+⋯or∑n=1∞(−1)n+1bn=b1−b2+b3−b4+⋯or
 ∑n=1∞(−1)nbn=−b1+b2−b3+b4−⋯∑n=1∞(−1)nbn=−b1+b2−b3+b4−⋯