Use Python to solve each problem. (NOTE: to plot sequences, we will be using matplotlib.pyplot rather than sympy.plot. Refer to the links in the help documentation if you are not familiar with pyplot.)

1. Given the sequence an = n n + 1 n + 3

(a) Evaluate a10, a20, a30, a40, a50 to numerically estimate the limit.

(b) Plot the first 50 points of the sequence to graphically estimate the limit.

(c) Rationalize an (multiply numerator and denominator separately), then take the limit. Confirm your answer by computing limn an directly.

2. Given the sequence an = ln 1 + 1 n

(a) Show an 0.

(b) Find the first 10 terms of the sequence, then the first 10 partial sums (NOTE: use for loops and symbolic computations, NOT arange and numeric computations, to see what is happening!).

(c) Plot the first 50 terms of the sequence and the first 50 partial sums on the same graph.

(d) From part (b), write an expression for sn and use it to show X n=1 an diverges.

3. (a) Use a while loop to create the recursive sequence a1 = 1, an+1 = 7 2 an until |an+1 an| < .00001 and print the resulting sequence to numerically show an is increasing and an < 7.

(b) Plot the sequence from part (a) to graphically show an is increasing and an < 7

(c) Assuming the statements in parts (a) and (b) are true, the limit exists, so let an = an+1 = L and solve for the exact limit.