Series

51 problems · Numerical series and power series.

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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.207Geometric Series

#geometric series #convergence #summation

Series

High school

Ex.208Geometric series and repeating decimals

#geometric series #decimals #sum

Find the sum

High school

Ex.209Geometric series with a parameter

#geometric series #convergence domain #parameters

Find the sum

High school

Ex.332Radius of convergence

#power series #ratio test

Power series

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.372Splitting the sum of 1/n² into even and odd terms

#absolute convergence #rearrangement #pi squared

Find the sum

Ex.385Series with factorials and the number e

#exponential series #factorials #e

Find the sum

Ex.388Telescoping series with partial fractions

#telescoping #partial fractions #sums

Find the sum

Ex.389Telescoping with logarithms

#telescoping #logarithm #divergence

Find the sum

Ex.390Telescoping with a shift p

#telescoping #partial fractions #harmonic numbers

Find the sum

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.772Power series of ln(1+x) and arctan x

#integration term by term #Leibniz series #ln 2

Power series

Ex.773Radius of convergence, four examples

#radius of convergence #ratio test #endpoints

Power series

Ex.774The sine of an alternating term

#alternating series #odd function #conditional convergence

Convergence tests

Ex.775Solving y″ = −y with a power series

#differential equations #power series #cosine #recurrence

Power series

Ex.776Summing power series in closed form

#power series #differentiation term by term #closed form

Power series

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.1393Three logarithmic series

#logarithmic series #telescoping #conditional convergence

Find the sum

Ex.1394Six numeric series: convergence study

#ratio test #root test #comparison #absolute convergence

Convergence tests

Ex.1395Three convergent series with explicit sums

#geometric series #telescoping #arctangent identity #d'Alembert

Find the sum

Ex.1396Three series with computed sums

#geometric series #telescoping #arctan identity

Find the sum

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.1490The generating function of Fibonacci numbers

#generating function #Fibonacci #partial fractions #radius of convergence

Power series

Ex.1499A series of functions

#series of functions #normal convergence #pointwise convergence

Power series

Ex.1506The generating function of central binomial coefficients

#binomial series #generalised binomial coefficients #radius of convergence

Power series

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

#ratio test #central binomial coefficients #induction #Stirling

Convergence tests

Ex.1510The Euler-Mascheroni constant as a series

#Euler-Mascheroni constant #harmonic numbers #telescoping #logarithm

Find the sum

Ex.1511e^x e^y = e^(x+y) from power series

#Cauchy product #exponential series #binomial theorem

Power series

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.2074Rearranging a conditionally convergent series

#conditional convergence #rearrangement #Riemann theorem #ln 2

Find the sum

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