Maths WorkshopMaths problems with full solutions

Convergence tests

29 problems · Comparison, ratio test and other criteria.

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Ex.9Terms that do not tend to zero

#divergence test #necessary condition

Convergence tests

Ex.24Equivalents, and a sign trap

#equivalents #comparison test #positive series

Convergence tests

Ex.358A square root whose difference tends to 1/2

#divergence test #square roots #asymptotics

Convergence tests

Ex.361The Bertrand series, all cases

#Bertrand series #parameters #integral test #comparison

Convergence tests

Ex.391Telescoping with square roots

#telescoping #square roots #divergence #p-series

Convergence tests

Ex.393The comparison test in practice

#comparison test #p-series

Convergence tests

Ex.397The ratio test

#ratio test #d'Alembert #factorials

Convergence tests

Ex.662Equivalence between $\sum u_n$ and $\sum\ln(1+u_n)$

#equivalence #logarithm #ratio test #convergence

Convergence tests

Ex.700Convergence of the series of 1/n²

#series #comparison

Convergence tests

Ex.702Divergence of the harmonic series

#series #divergence

Convergence tests

Ex.724Cancellation revealed by a Taylor expansion

#Taylor expansion #equivalents #cancellation

Convergence tests

Ex.725A telescoping sum that proves ζ(2) converges

#telescoping #comparison #zeta function

Convergence tests

Ex.726Absolute, conditional or no convergence

#absolute convergence #alternating series #p-series

Convergence tests

Ex.774The sine of an alternating term

#alternating series #odd function #conditional convergence

Convergence tests

Ex.779Alternating series and the error bound

#alternating series test #remainder #error estimate

Convergence tests

Ex.785The integral test and Bertrand series

#integral test #Bertrand series #logarithm

Convergence tests

Ex.787The limit comparison test

#limit comparison #equivalents #Taylor expansion

Convergence tests

Ex.791The root test

#root test #Cauchy #exponential limits

Convergence tests

Ex.793Two alternating series with logarithms

#alternating series test #monotonicity #conditional convergence

Convergence tests

Ex.1394Six numeric series: convergence study

#ratio test #root test #comparison #absolute convergence

Convergence tests

Ex.1470The series of 1/(ln n)^(ln n)

#comparison test #exponential form #logarithm

Convergence tests

Ex.1473Cauchy's condensation test

#condensation test #p-series #monotone terms

Convergence tests

Ex.1474A Cauchy product that diverges

#Cauchy product #conditional convergence #Mertens theorem

Convergence tests

Ex.1508A ratio test that fails, and how to conclude

#ratio test #central binomial coefficients #induction #Stirling

Convergence tests

Ex.1515The ratio test with a parameter

#ratio test #Stirling formula #critical case

Convergence tests

Ex.2034Four series involving logarithms

#root test #comparison #equivalent #logarithm

Convergence tests

Ex.2072Convergence of the series of sin(n)/n

#Abel summation #Dirichlet test #trigonometric series

Convergence tests

Ex.2081Reciprocals of integers with no digit 9

#comparison test #counting #harmonic series #Kempner

Convergence tests

Ex.2082A series with an equivalent that converges but does not

#equivalents #alternating series #counterexample

Convergence tests