Re: Limit of a sequence

On Wed, 16 Apr 2008 00:58:01 EDT, TCL <tlim1@xxxxxxx> wrote:

In article
<15986032.1208228781859.JavaMail.jakarta@xxxxxxxxxxxxx
forum.org>,
TCL <tlim1@xxxxxxx> wrote:
For positive integers n, define

s_n = sum_{k=1 to n} of (1/k)* (1-(1-1/n)^k).

Prove that s_n --> 1 as n-->infty.


n
--- n k 1
> - ( 1 - (1 - 1/n) ) -
--- k n
k=1

is a Riemann Sum for

|\1 1 -x
| - ( 1 - e ) dx
\| 0 x

with x = k/n.

As noted by WWW, this is not quite a Riemann sum. To justify your steps, one needs to prove

Sum_{k=1 to n} (1/k)* ( exp(-k/n) - (1-1/n)^k )

converge to 0 as n-->infty. The fact that lim (1-1/n)

^n=e^{-1} may be helpful, but that does not prove that

the above limit is 0.

True. But a complete proof is not hard by Dominated
Convergence, using the fact that (1-1/n)^n < 1/e

TCL

David C. Ullrich
.

Relevant Pages

  • Re: Limit of a sequence
    ... TCL wrote: ... with x = k/n. ... This integral can be reduced using power series to ...
    (sci.math)
  • Re: Limit of a sequence
    ... TCL wrote: ... with x = k/n. ... this is not quite a Riemann sum. ... take out the trash before replying ...
    (sci.math)
  • Re: Limit of a sequence
    ... TCL wrote: ... this is not quite a Riemann sum. ... To build an epsilon-delta proof we will ... Rob Johnson ...
    (sci.math)
  • Re: Limit of a sequence
    ... David C. Ullrich wrote: ... TCL wrote: ... with x = k/n. ... this is not quite a Riemann sum. ...
    (sci.math)
  • Re: Limit of a sequence
    ... TCL wrote: ... with x = k/n. ... this is not quite a Riemann sum. ... Exponentiating back, using e^x = 1 + Ofor x bounded gives ...
    (sci.math)
Quantcast