Expert: Paul Klarreich - 5/26/2007

Question
QUESTION: hello Paul,

i'm studying calculs 102, and i have a 2nd hour exam on sunday, and i need some help with some questions.

- which of the sequences converge and which diverge? find the limit of each convergent sequences.

- a(n) = ((-1)^(n+1))/(2n-1)
- a(n) = (ln(n+1))/sqrt(n)
- a(n) = (1 - (1/(n^2))^n

Thanks for the help

ANSWER: Questioner:   Ibrahim
Category:  Calculus
Subject:  infinite sequences
Question:  hello Paul,

i'm studying calculs 102, and i have a 2nd hour exam on sunday, and i need some help with some questions.

- which of the sequences converge and which diverge? find the limit of each convergent sequence.

- a(n) = ((-1)^(n+1))/(2n-1)
- a(n) = (ln(n+1))/sqrt(n)
- a(n) = (1 - (1/(n^2))^n

Thanks for the help
..........................................
Hi, Ibrahim,

I assume that since you said 'sequences' you really mean 'sequences' and not series.

((-1)^(n+1))/(2n-1)  is a fraction:

(-1)^(n+1)
----------
 2n - 1

Since the denominator is approaching infinity while the absolute value of the numerator is always equal to 1, this sequence converges to zero.

...........................
(ln(n+1))/sqrt(n)  is also a fraction.  It is well known that ln(n) grows slower than any power of n, so this will also converge to zero.  Need a proof?  Use l'Hospital's rule:
        ln(n+1)
lim     --------- =
n->inf    n^1/2

         1/(n+1)
lim     ----------- =
n->inf   (1/2)n^-1/2

         1/(n+1)
lim     ------------- =
n->inf   1/(2 n^1/2)

         2 n^1/2
lim     ------------- =
n->inf     (n+1)

         2
lim     ----------------- =
n->inf   n^1/2 + 1/n^1/2
      2
------------- = 0
infinity + 0
..........................................
- a(n) = (1 - (1/(n^2))^n

There are two well-known limits related to this:

lim    (1 + 1/h)^h  = e
h->inf

lim    (1 - 1/h)^h  = 1/e
h->inf


(1 - (1/(n^2))^n = [ (1 - (1/(n^2))^(n^2) ]^1/n

Let  h = n^2.  Then that is:

lim     [ (1 - 1/h)^h  ]^1/sqrt(h) = (1/e)^0 = 1
h->inf


---------- FOLLOW-UP ----------

QUESTION: Hello,

can u please tell me what applies to sequences and series?! and wut only applies to one of them? like rules and theorms?

Answer
Followup:

A sequence is a LIST: a function whose domain is the positive integers.  So when we write

A[7} = (whatever)

we are referring to a function A(x) whose value at  x = 7 is the whatever, and that there will be no such thing as  A(3.7),  or  A(-5), because the domain of A is positive integers.  [Well, it could include zero, I suppose.]

A SERIES is also a list, but it is based on some other sequence and it is the sequence of PARTIAL SUMS.  So, for example, if the

Sequence  A[n] is defined by  A[n] = 1/n,

then the

Series:  S[n] is defined as  S[n] = 1/1 + 1/2 + 1/3 + ... + 1/n

and the Partial Sum

A[7] = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7

and it is common to use the SUMMATION symbol (which I cannot make here) to indicate it, as in:
         n
S[n] = SUMMATION (1/k)
        k = 1

For theorems regarding convergence, see chapter 12 of your standard calculus text.