I have been thinking about this problem:(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Convergent Series

**Physics Forums | Science Articles, Homework Help, Discussion**