Does a uniformly convergent sequence imply a convergent series

In summary, a convergent sequence is a series of numbers that approaches a single limit as the number of terms increases, while a convergent series is the sum of all the terms in a sequence. A uniformly convergent sequence can have a divergent series because uniform convergence only guarantees that the sequence of partial sums will converge, not the actual series. To determine if a uniformly convergent sequence has a convergent series, you can check if the sequence of partial sums is bounded. It is possible for a non-uniformly convergent sequence to have a convergent series, as non-uniform convergence does not guarantee the convergence of the sequence of partial sums. And finally, a series can converge but not be uniformly convergent, as uniform convergence
  • #1
gottfried
119
0
Does anybody know if this statement is true?

[itex]\sum[/itex] fn converges absolutely and uniformly on S if ( fn) converges uniformly.

Also if R is the radius of convergence and |x|< R does this imply uniform and absolute convergence or just absolute convergence.
 
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  • #2
Suppose fn converges uniformly to something other than 0.

The phrase "radius of convergence" only applies to power series and, in that case, the series converges both absolutely and uniformly inside the radius of convergence.
 

Related to Does a uniformly convergent sequence imply a convergent series

1. What is the difference between a convergent sequence and a convergent series?

A convergent sequence is a series of numbers that approaches a single limit as the number of terms increases. A convergent series is the sum of all the terms in a sequence. So while a convergent sequence may approach a limit, a convergent series may or may not have a finite sum.

2. Can a uniformly convergent sequence have a divergent series?

Yes, a uniformly convergent sequence can have a divergent series. This is because uniform convergence only guarantees that the sequence of partial sums will converge, not the actual series itself. The series can still diverge if the limit of the partial sums goes to infinity.

3. How can I determine if a uniformly convergent sequence has a convergent series?

You can determine if a uniformly convergent sequence has a convergent series by checking if the sequence of partial sums is bounded. If the partial sums are bounded, then the series will converge. However, if the partial sums are unbounded, then the series will diverge.

4. Is it possible for a non-uniformly convergent sequence to have a convergent series?

Yes, it is possible for a non-uniformly convergent sequence to have a convergent series. This is because non-uniform convergence does not guarantee that the sequence of partial sums will converge, but it is still possible for the series to converge if the partial sums happen to converge.

5. Can a series converge but not be uniformly convergent?

Yes, a series can converge but not be uniformly convergent. This is because uniform convergence is a stronger condition than convergence. A series can still converge if the terms approach zero fast enough, even if they do not approach zero uniformly. However, if a series is uniformly convergent, then it must also be convergent.

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