Given f_n(x) = (x^2)/(1+x^2)^n, find the region of convergence. I'm not sure what the "region" is. Is it just the same as the radius of convergence?
The 'infinite sum' of functions...
(1)
... does converge for all real values of the variable. For is...
(2)
... and the (1) is a convergent 'geometric series'. For every term of (1) vanishes so that the 'infinite sum' vanishes also...
For the complex 'infinite sum' of functions...
(3)
... the problem is a little more 'complex'... of course ...
Merry Christmas from Italy
I don't really understand your question.
Is this an infinite series?
Is it supposed to be, for example
?
If it is, you're trying to find the values for which the function converges.
Note that this can be rewritten as
.
Can you see that it is a geometric series, with and ?
Geometric series converge when .
So this function converges when
Solve for .
A "power series", a series of the form has a "radius of convergence" but other kinds of series may not.
In particular, you can think of the series you are given, as a geometric series, with and .
That's chisigma's point.
Now, you have probably learned, long ago, that the geometric series will converge as long as |r|< 1. That's where chisigma gets . In fact, since is never negative, which is equivalent to or . That's true for all x except ___ and the region of convergence is "all x except x= ___".