Evaluate Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not started infinite series.

Series Problem

Problem: Show that the convergence of a series is not affected by changing a finite number of its terms. (of course, the sum may well be changed).

Series Proof

Problem: Show that if a convergent series of real numbers contains only a finite number of negative terms, then it is absolutely convergent.

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive on the function

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive: a) the function is a Lipschitz function with a Lipschitz constant L in (0,1) or b) the absolute value of f'(x) is less than one for all x in [a,b]

I have a function, with domain (a,b) --> R, which is monotone (which impies it is bounded at each constant in (a,b)). I have two conditions that I am trying to meet

First, I am looking for an example of a monotone function with (a,b)-->R that is unbounded and then I need to verify that the function has lim_x-->c^+ less than or equal to Lim_x-->d^- whenever a < c < d < b