Maths WorkshopMaths problems with full solutions

Sequences

32 problems · Convergence of real sequences and recursive sequences.

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Ex.2A recurrence and a proof by induction

#induction #recurrence #closed form

Ex.82The ε–N definition of a limit

#definition of a limit #epsilon-N #divergence

Ex.198A telescoping sum

#telescoping #series #partial sums

Ex.201Arithmetic and geometric sequences

#arithmetic sequence #geometric sequence #sums

Ex.212The sum of the first n squares

#induction #sums #limits

Ex.350The n-th root of n

#sequences #logarithm #growth comparison

Ex.355A linear recurrence of order two

#linear recurrence #characteristic equation

Ex.356A telescoping product

#telescoping #products #factoring

Ex.357A telescoping sum with factorials

#telescoping #factorials

Ex.359An arithmetico-geometric sequence

#recurrence #fixed point #geometric sequence

Ex.375Three limits by the squeeze theorem

#squeeze theorem #sums #n-th roots

Ex.698A recursive sequence converging to 2

#sequences #monotone convergence

Ex.722A recurrent sequence and its fixed point

#fixed point #contraction #mean value theorem

Ex.730Binet's formula for the Fibonacci numbers

#Fibonacci #golden ratio #linear recurrence #limits

Ex.735Convergence to the golden ratio

#golden ratio #contraction #fixed point

Ex.761Heron's method for √2

#Newton's method #quadratic convergence #square roots

Ex.777Supremum, infimum, limsup and liminf

#supremum #limsup #liminf #bounded sequences

Ex.784The hierarchy of growth

#growth comparison #ratio test #factorial

Ex.792Three cases of order-two recurrences

#linear recurrence #double root #complex roots

Ex.1468A recurrence with a double fixed point

#fixed points #monotone sequences #parameter dependence

Ex.1471Adjacent sequences and the irrationality of e

#adjacent sequences #number e #irrationality

Ex.1502The sequence sin(n) does not converge

#divergence #trigonometry #subsequences #contradiction

Ex.1503Speed of convergence

#speed of convergence #fixed point #mean value theorem

Ex.1504The alternating harmonic sums converge

#adjacent subsequences #alternating series #ln 2

Ex.1505The arithmetic–geometric mean

#adjacent sequences #AM-GM #means

Ex.1507The Cauchy criterion

#Cauchy sequences #completeness #series

Ex.1509The Cesàro lemma

#Cesaro mean #epsilon-N #averages

Ex.1514From the ratio to the n-th root

#Cesaro lemma #n-th root #ratio #Stirling

Ex.1516Monotonicity of (1 + 1/n)^n

#number e #AM-GM inequality #monotone convergence

Ex.2069The root of x^n + x = 1

#implicit sequences #monotonicity #intermediate value theorem

Ex.2073Interlacing an arithmetic and a geometric progression

#arithmetic progression #geometric progression #inequalities #construction

Ex.2079The logistic sequence decays like 1/n

#Cesaro lemma #asymptotic equivalent #recurrence