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## Homework Statement

I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent.

## Homework Equations

The relevant equation used is that lim (n-> infinity) sup|(fn(x) - f(x))|= 0 implies uniform convergent.---- (1)

## The Attempt at a Solution

It is obvious lim (n-> infinity) fn(x) = f(x) = 0 for x not equal to zero. And when x=0, fn(0) = 0 and hence as n-> infinity, fn(0) = f(0) =0.

As mentioned in Ross and ( I can see) that fn(x) is pointwise convergent. But, it looks like for all x, the function fn(x) converges to f(x)=0. So, I am unclear as to why it is not uniform convergent? Please clarify.

The book further uses the theorem (1) to prove uniform converfence, which I can understand.

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