A quick R code demonstrating some of the logic behind sequences and convergence of sequences.

sequences-and-convergence.R

### Sequences and Convergence
### Jane Lawrence Sumner
### POLS 508
### Dec. 15, 2015

##### Here are the functions that make the thing work. Scroll down for examples.
fxn <- function(x){
  fx <- (x+1)/x  ## function goes here
  return(fx)
}


## this function generates the nth value of the sequence
sequence <- function(n){
  ans <- NULL
  ns <- seq(from=1,to=n,by=1)
  for(i in 1:length(ns)){
    ans <- c(ans,fxn(ns[i]))
  }
  #plot(c(1:n),ans,type="l")     ## it'll plot it if you uncomment this
  return(ans)
}

converges <- function(){        ## this function has no input and tells you whether the 
  ## sequence converges after 1000 iterations (roughly)
  yn <- round(sequence(1000)[999],2)==round(sequence(1000)[1000],2)
  if(yn==T){
    return(round(sequence(1000)[1000],2))
  } else{ 
    return("Does not converge.")
  }
}

# this function tells you, for a given epsilon>0, what is your N such that n>N
# implies d(x^n,x)<epsilon. (Recall from your notes, though, that to prove convergence
# this must be true for an epsilon>0, not just a given epsilon>0)
how.many <- function(epsilon,limit){
  if(converges()=="Does not converge."){
    return("This sequence does not converge.")
  } else{
    is.limit <- round(sequence(1000)[1000],2)==limit
    if(is.limit==T){
      n <- 1
      while(abs(limit-fxn(n+1))>=epsilon){
        n <- n+1
      }
      beforethat <- abs(limit-fxn(n-1))
      afterthat <- abs(limit-fxn(n+1))
      return(list(N=n))
    }
    else{
      return(paste("This sequence does not converge to ",limit,".",sep=""))
    }
  }
}



#######################
## FOR INSTANCE
#######################

# 1 - Sequence s(n)=n+3
fxn <- function(x){
  fx <- x+3  
  return(fx)
}

sequence(5)
converges()    ## This sequence does not converge.
how.many(.02,5) ## What is N such that n>N -> metric distance less than epsilon?
# Nope. Sequence does not converge.



# 2 - Sequence s(n)=(n+1)/n
fxn <- function(x){
  fx <- (x+1)/x
  return(fx)
}

sequence(5)    # First five elements of the sequence.
converges()    ## This sequence does converge. It converges to 1.
how.many(.02,5) ## What is N such that the sequence converges to 5 at an epsilon of .02?
how.many(.02,1) ## N=50, so for n>50, metric distance is less than .02
how.many(.001,1) ## N=999, so for n>999, metric distance is less than .02
sequence(1002)-1<.001  ## Can confirm by looking at distances for each n.
which(sequence(1002)-1<.001)  ## Yes, n>N -> metric distance less than epsilon



how.many(.1,1) ## N=10, so for n>10, metric distance is less than .1
sequence(12)-1<.1  ## Can confirm by looking at distances for each n.
which(sequence(12)-1<.1)  ## Yes, n>N -> metric distance less than epsilon