Math Real Analysis

MATH 3001 W21 FINAL EXAM

:

4) Find all a >o such that the following tones converges :

C-opt] z lnfn

"

)

NII qh

4) Suppose that the series Ian converges conditionally and h? I

A-opt] set pn= ; ( lannan ) . Show that Inpn diverges .

(3) For a>o, decide if the following genes is absolutely convergent, convergent or divergent :

G-opt]

n

C- yh ( lnnjlhh ya

-

(4) Let fnlxj = n fn It'm

' hx ,

✗>0 .

Ppt] (a) Find the pointwise limit function f-1×1 .

Apt] f) Does fn converge to f uniformly on lo , a) ?

(5) Consider 1--1×1=2%1×1 4-pt) (a) Find all ✗ER so that the series converges .

[b-pt) (b) Show that FIN is continuous for ✗ c- C-1,1 ) .

(6) Use the Dirichlet test to show that Fogh é "

✓ht✗T Copt converges uniformly on [90) .

(7) Find the Taylor series of f-1×1= h×

✗ I ✗ 20 , about

the point c-1 . ["Pt)

what is the radius of convergence ?

Hotel : 70 points)