Calculus 2 - Mid Semester Exam

1.     Calculate the sum of the following series

2. Calculate the radius of convergence of the following series and their sum

3.    Test whether the following sequences uniformly converge

4. Evaluate

5.     Calculate the power series of the following functions

6.    Consider the following numerical sequence given by

            Study the convergence of

            Deduce that:

            Using:

            calculate k

7.        Let p be an integer. Choose a family of positive integers .

           For each integer n, we call:

            Calculate the sum

8.         Denote

            .

            Show that the function is defined over R

            Calculate

            Using the previous result, show that

            Deduce an equivalent for f(x) as x goes to infinity

9.        (Riemann's Zeta Function)

            Show that

            Calculate

            Show that:

            Deduce that ζ converges in infinity

            Show that in s=1,

            Show the following equality (first proven by Euler)

                Deduce that diverges

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