please i need a formal proof of comparison test for improper integrals..

Hint: Use the fact that an improper integral is a limit of a proper integral as one or both bounds approach either ±∞ or values where the function is discontinuous. Or could be that a value inside an interval where the function is discontinuous.

If you also want to start by proving of the comparison test for proper integrals, you can do so by interpreting a definite integral as a limit of a Riemann sum.

I hope this helps.

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Added in reply to comment below:

Here is an outline. You will need to fill in the details since that will give you an opportunity to practice writing formal proofs.

First you prove a lemma for proper integrals since an improper integral is a limit of a proper integral.

Lemma:

If f(x) and g(x) are continuous in a finite interval [a,b]

with f(x) ≥ g(x) ≥ 0 then ∫_{a}^{b} f(x) dx ≥ ∫_{a}^{b} g(x) dx.

Proof: Divide the interval [a,b] into n subintervals I_{k} of equal lengthfor k = 1,...,n, and pick an x_{k} from the k^{th
}interval.

Then the integrals can be approximated by Riemann sums,

Sum[|I_{k}|*f(x_{k})] and Sum[|I_{k}|*g(x_{k})].

Term by term comparison shows that Sum[|I_{k}|*f(x_{k})] ≥ Sum[|I_{k}|*g(x_{k})].

Letting n → ∞, the sums approach the integrals and the lemma follows

Proof of the comparison test:

Let f(x) ≥ g(x) ≥ 0 over (a,b) wherever both f and g are continuous. Here a could be -∞, and b could be +∞.

Also assume there may be x values where f or g is discontinuous and let them partition (a,b) into intervals I_{k} = (x_{k-1}, x_{k}).

Let F_{k} = ∫_{I}_{k} f(x) dx and G_{k} = ∫_{Ik} g(x) dx.

Then for any subinterval [c,d] of I_{k}, our lemma guarantees

∫_{c}^{d} f(x) dx ≥ ∫_{c}^{d} g(x) dx.

Taking the limit c → x_{k-1} and d → x_{k} gives that F_{k} ≥ G_{k}

Since ∫_{a}^{b} f(x) dx = Sum(F_{k}) and ∫_{a}^{b} g(x) dx = Sum(G_{k}), you get that

∫_{a}^{b} f(x) dx ≥ ∫_{a}^{b} g(x) dx.

Finally, using this inequality, if the integral in f converges, so does the one in g, while if the one in g diverges, so does the one in f.

## Comments

but i need a complete proof:s.. I need to submit my assignment after two days and i am unable to do the proof:s