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Sketch of Solution to problem 9:
Given that a0 = 0, a1 = 1, and
an1 + an2
, n 2.
an =
2
From the above recurrence relation, we have n 2
2an = an1 + an2 ,
which implies that
2(an an1 ) = (an1 an2 ).
In other words
(an1 an2 )
.
2
Doing it successively, we g

Department of Mathematics, Indian Institute of Science
Analysis I (MA 221) Test 1
August 2011
Instructor: A.K.Nandakumaran September 24, 9.30 - 11.30 am.
Maximum 25 marks. Write your answers clearly, DO NOT use short notations for English
words.
1.
Recall

GENERAL
I ARTICLE
The Meaning of Integration
A](
1. Riemann
- II
Nandakumaran
Integration
In Part I, we defined the area under a curve, which is
given by a continuous function, as the integral of such
a function.
In this part, we ~iscuss the definition of

GENERAL
I ARTICLE
The Meaning of Integration - I
A K Nandakumaran
1. Introduction
Integration is the 'inverse of differentiation' is what we
are told in our beginning calculus course. It is indeed the
case when the given function is continuousl. Augustin

Department of Mathematics, IISc., Bangalore 560012
ANALYSIS I (MA 221, 3:0, August 2011)
Exercise 3
A. K. Nandakumaran
Submit before
1. Let
f (x) =
2
xn e1/x , if x > 0
0 at x = 0
1
2
e1/x exists for any polynomial Pn .
x0 Pn (x)
Show that lim f (x) exist

Department of Mathematics, IISc., Bangalore 560012
ANALYSIS I (MA 221, 3:0, August 2011)
Exercise 12
A. K. Nandakumaran
Submit before
P
P
(1) Define d1 (x, y) = ni=1 |xi yi |, d2 (x, y) = ( ni=1 |xi yi |2 )1/2 ,
d (x, y) = max |xi yi |.
i=1, ,n
(2)
(3)
(4

Department of Mathematics, IISc., Bangalore 560012
ANALYSIS I (MA 221, 3:0, August 2011)
Exercise 1
A. K. Nandakumaran
Submit before
(1) a). Prove Euclids algorithm using mathematical induction:
Given m, n N, m 6= 0, there exist q, r N such that
n = mq +