5CCM221a Analysis I Summer 2018
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5CCM221a Analysis I
Summer 2018
Section A
1. (15 points) Let (an )neN be a sequence in R and a e R.
(i) (5 points) Define the notion of convergence and lim an = a.
(ii) (5 points) Using the definition of convergence, show that lim 1 = 0.
(iii) (5 points) Compute lim
2. (20 points) Let (an )neN be a sequence in R and a e R.
o o
(i) (5 points) Define the notion of convergence of an and an = a. n=1 n=1
(ii) (5 points) Show that 
is convergent and compute its limit.
o xn
n
n=1
3. (15 points) Let I S R be an interval and f : I → R.
(i) (5 points) Define what it means for f to be continuous. (ii) (10 points) Let f be Lipschitz continuous, i.e.,
3L e R>0 Ax, y e I : |f (x) - f (y)| < L |x - y| . Show that f is continuous using the definition of continuity.
Section B
4. (25 points) Let I S R be an open interval and f : I → R.
(i) (5 points) Define what it means for f to be differentiable on I and define the derivative f/ : I → R of f .
(ii) (10 points) Give an example of a function g that is continuous but not differentiable. Prove continuity and non-differentiability.
(iii) (10 points) Let f e C1 (I) and a, b e I such that a < b. Show that f is Lipschitz continuous on [a, b], i.e.,
3L e R>0 Ax, y e [a, b] : |f (x) - f (y)| < L |x - y| .
5. (25 points)
(i) (20 points) State and prove the Fundamental Theorem of Calculus. (ii) (5 points) State and prove the integration by parts formula.
6. (25 points) Let I S R be an open interval and f : I → R.
(i) (5 points) Define what it means for f to be analytic. (ii) (20 points) Let I = R and f (x) := x sin(x).
(a) (10 points) Compute all derivatives of f and show that its Taylor
series at 0 is given by Tf,0(x) = x2n .
n=1
(b) (10 points) Show that f is analytic.
2022-07-28