Let f:[1,∞[ →R be a positive continuous decreasing function. Show if m,n ∈N and m<n that the integral

∫ f(x) dk from (m+1) to (n+1) ≤∑ f(k) (sum from k=1 to n) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to n)

Problem 2 :

From problem 1 prove if ∫ f(x) dk on the interval from 1 to infinity is convergent that the integral ∫ f(x) dk from (m+1) to (infinity) ≤∑ f(k) (sum from k=1 to infinity) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to infinity)

Problem 3 :

Let snm = =∑ 1/k (sum from k=1 to nm) - (sum from k=1 to n times m)

Show from problem 1 if sn=∑ 1/k (sum from k=1 to n) for all n ∈N that the sum difference limit : lim (snm - sm) = ln(n)

Problem 4 :

Proff that the functional serie fn given by fn : R→ R for n=1,2,3 ..... given by

fn(x) = ____1________

1+ exp((n(2-x))

is pointwise convergent and find the limiting function.

Is the functional serie also uniform convergent on R ? Argue for your result.

Help proving these four problems is urgently needed.

Thanks

Pragma