Convergent Series

  • Thread starter BSCowboy
  • Start date
  • #1
31
0

Main Question or Discussion Point

I have been thinking about this problem:
Determine whether the following series are convergent in [tex]\left(C[0,1],||\cdot ||_{\infty}\right)[/tex] and [tex]\left(C[0,1],||\cdot ||_{1}\right)[/tex].
when
[tex]f_n(t)=\frac{t^n}{n}[/tex]

In the supremum norm, this seems pretty straightforward, but in the integral norm I am confused since,
[tex]\left\|\sum\frac{t^n}{n}\right\|_1\leq\sum\left\|\frac{t^n}{n}\right\|_1=\sum\int_0^1\frac{t^n}{n}dt=\sum\left[\frac{t^{n+1}}{n^2+n}\right]_0^1=\sum\frac{1}{n^2+n}<\sum\frac{1}{n^2} [/tex]
and, I think this converges as [tex]n\rightarrow\infty[/tex], but our instructor said this did not converge, or maybe I heard him incorrectly. So, does this converge? He asked us to show the series is Cauchy and that the limit is not in the space as well. What am I missing?
 
Last edited:

Answers and Replies

  • #2
mathman
Science Advisor
7,759
415
Could you be more precise as to what is supposed to be converging to what? Partial sums of fn(t) or fn(t) itself as a sequence.
 
  • #3
31
0
Actually, I think I might have figured it out:
[tex]f_n(t)=\frac{t^n}{n} \text{ in } \left(C[0,1],\|\cdot\|_1\right)[/tex]

[tex]\|f_n\|_1=\int_0^1f_n(t)dt=\frac{1}{n(n+1)} \text{ and } \sum_{n=1}^{\infty}\frac{1}{n(n+1)}\rightarrow 0[/tex]

But, because the space [tex]\left(C[0,1],\|\cdot\|_1\right)[/tex] is not complete all we know is that [tex]f_n(t)[/tex] is Cauchy. I was missing the part about the space being complete.
Also, you can see that limit is not an element of C[0,1].
 
  • #4
statdad
Homework Helper
1,495
35
[tex]
\sum_{n=1}^\infty {\frac 1 {n(n+1)} \not \to 0
[/tex]

but

[tex]
\lim_{n \to \infty} \frac 1 {n(n+1)} \to 0
[/tex]
 
  • #5
534
1
Also [tex]\sum_{n=1}^\infty \frac{1}{n(n+1)} = \sum_{n=1}^\infty \left[ \frac{1}{n} - \frac{1}{n+1} \right] = 1[/tex]
 

Related Threads for: Convergent Series

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
549
  • Last Post
Replies
1
Views
2K
Top