Sample final questions-

1. Consider the power series

_n=1∑^∞xⁿ/n³,          _n=1∑^∞x³ⁿ/2n,           _n=0∑^∞x^2n!.

For each power series, determine its radius of convergence R. By considering the series at x = ±R, determine the exact interval of convergence.

2. (a) Prove by using the definition of convergence only, without using limit theorems, that if (s_n) is a sequence converging to s, then lim_n→∞ s_n² = s².

(b) Prove by using the definition of continuity, or by using the ε-δ property, that f(x) = x² is a continuous function on R.

3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose r, s, t ∈ [0, 1] are defined so that r < s < t and f(r) = f(s) = f(t) = 0. Prove that there exists an x ∈ (0, 1) such that f''(x) = 0.

4. (a) Suppose that _n=0∑^∞aⁿ is a convergent series. Define a sequence (b_n) according to b_n = a_2n + a_2n+1 for n ∈ N ∪ {0}. Prove that _n=0∑^∞b_n converges.

(b) Construct an example of a series _n=0∑^∞a_n that diverges, but that if (b_n) is defined as above, then _n=0∑^∞ b_n converges.

5. Suppose f a is real-valued continuous function on R and that f(a)f(b) < 0 for some a, b ∈ R where a < b. Prove that there exists an x ∈ (a, b) such that f(x) = 0.

6. Let f be a real-valued function defined on an interval [0, b] as

Consider a partition P = {0 = t₀ < t₁ < . . . < t_n = b}. What are the upper and lower Darboux sums U(f, P) and L(f, P)? Is f integrable on [0, b]?

7. Let f be a decreasing function defined on [1, ∞), where f(x) ≥ 0 for all x ∈ [1, ∞). Prove that ₁∫^∞f(x)dx converges if and only if _n=1∑^∞f(n) converges.

8. Consider the function defined for x, y ∈ R as

(a) Prove that d defines a metric on R.

(b) What is the neighborhood of radius 1/2 centered on 0?

(c) Consider an arbitrary set S ⊆ R. Is S open? Is S compact?

9. Let x = (x₁, x₂) and y = (y₁, y₂) be in R². Consider the function

d(x, y) = |x₁ - y₁| + |x₂ - y₂|.

(a) Prove that d is a metric on R².

(b) Compute and sketch the neighborhood of radius 1 at (0, 0).

10. Consider a function f defined on R which satisfies

|f(x) - f(y)| ≤ (x - y)²

for all x, y ∈ R. Prove that f is a constant function.

11. Suppose that f is differentiable on R, and that 2 ≤ f'(x) ≤ 3 for x ∈ R. If f(0) = 0, prove that 2x ≤ f(x) ≤ 3x for all x ≥ 0.

12. Show that if f is integrable on [a, b], then f is integrable on every interval [c, d] ⊆ [a, b].

13. (a) Suppose r is irrational. Prove that r^1/3 and r + 1 are irrational also.

(b) Prove that (5 + √2)^1/3 + 1 is irrational.

14. By using L'Hopital's rule, or otherwise, evaluate

limx→0(x/1 - e^{-x^2-3x}),   lim_x→0(1/sin x - 1/x),       limx→0(x³/sin x - x).

15. Let a ∈ R. Consider the sequence (sn) defined as

Compute lim sup s_n and lim infs_n. For what value of a does (s_n) converge?

16. Consider the function f: R² → R defined as

f(x₁, x₂) = 1/x₁² + x₂² + 1.

With respect to the usual Euclidean metrics on R and R², prove that f is continuous at (0, 0) and at (0, 1).

17. (a) Calculate the improper integral

₀∫¹x^-p dx

for the cases when 0 < p < 1 and p > 1.

(b) Prove that

₀∫^∞x^-p dx = ∞

for all p ∈ (0, ∞).

18. Prove that if f is integrable on [a, b], then

lim_{d→b^- a}∫^d f(x) dx = _a∫^bf(x) dx.

19. Let f(x) = x², and define a sequence (s_n) according to s₁ = λ and s_n+1 = f(s_n) for n ∈ N. Prove that (s_n) converges for λ ∈ [-1, 1], and diverges for |λ| > 1.

20. Consider the three sets

A = [0, √2] ∩ Q,                                 B = {x² + x - 1: x ∈ R},                      C = {x ∈ R: x² + x - 1 < 0}.

For each set, determine its maximum and minimum if they exist. For each set, determine its supremum and infimum. Detailed proofs are not required, but you should justify your answers.

21. Let f_n(x) = x - xⁿ on [0, 1] for n ∈ N.

(a) Prove that f_n converges pointwise to a limit f, and determine f.

(b) Prove that f_n does not converge uniformly to f.

(c) Find an interval I contained in [0, 1] on which fn → f uniformly.

(d) Prove that the f_n are integrable, that f is integrable, and that ₀∫¹f_n → ₀∫¹f.

22. Define

for x ∈ R.

(a) Calculate F(x) = ₀∫^xf(t)dt for x ∈ R.

(b) Sketch f and F.

(c) Compute F' and state the precise range over which F' exists. You may make use of the second Fundamental Theorem of Calculus.

23. (a) Let f and g be continuous functions on [a, b] such that _a∫^b f = _a∫^bg. Prove that there exists an x ∈ [a, b] such that f(x) = g(x).

(b) Construct an example of integrable functions f and g on [a, b] where _a∫^b f = _a∫^b g but that f(x) ≠ g(x) for all x ∈ [a, b].

24. Define the sequence of functions hn on R according to

(a) Sketch h₁, h₂, and h₃.

(b) Prove that h_n converges pointwise to 0 on R/{0}. Prove that lim_n→∞ h_n(0) = ∞.

(c) Let f be a continuous real-valued function on R. Prove that lim_n→∞-∞∫^∞h_n f = f(0).

(d) Construct an example of an integrable function g on R where lim_{n→∞ -∞}∫^∞ h_ng, exists and is a real number, but does not equal g(0).

25. Consider the function

f(x) = x/1 + x.

on the interval [0, ∞).

(a) Show that lim_x→∞ f(x) = 1, and that 0 ≤ f(x) < 1 for all x ∈ [0, ∞).

(b) Sketch f.

(c) Calculate f', f'', and use them to construct the partial Taylor series at x = 1 with the form

f_T(x) = _n=0∑²((x - 1)ⁿf⁽ⁿ⁾(1)/n!).

(d) Show that f_T can be written as a quadratic equation with the form ax² + bx + c, and compute a, b, and c.

(e) Add a sketch of f_T to the sketch of f. [Note: f_T(1) = f(1) so the two curves should intersect at x = 1.]