Page 265 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

This gives us the following method for finding the limit of sum of n terms of a series: Notes
1. Write the general rth term of the series.

r
1 æ ö 1 r
,
2. Express it as f ç ÷ the product of and a function of .
n è ø n n
n
r 1
3. Change to x and to dx and integrate between the limits 0 and a. The n value of the
n n
resulting integral gives the limit of the sum of n terms of the series.
Since each term of a convergent series tends to 0, the addition or deletion of a finite number of
terms of the series does not affect the value of the limit. Similarly, you can verify that
2n 1 æ r ù ö 2
é
lim å ê f ç ÷ ú =  f (x) dx,
n¥ r 1 n è n ø û 0
= ë
é
3n 1 æ r ù ö 3
lim å ê f ç ÷ ú =  f (x) dx, and so on.
n¥ r 1 n è n ø û
= ë
As an illustration of these results, consider the following examples.

Example: Find the limit, when n tends to infinity, of the series

1 1 1 1
+ + +¼+ .
n 1 n + 2 n + 3 n + n
+
æ ö
n 1 n 1 ç 1 ÷
Solution: General (rth) term of the series is å = å .
r 1 n r+ r 1 n ç r ÷
=
=
ç 1 + ÷
è n ø
æ ö
n 1 ç 1 ÷ 1 1
Hence, lim å =  = dx = log 2.
n¥ r 1 n ç r ÷ 0 1 x+
=
ç 1 + ÷
è n ø
Example: Find the limit, when n tends to infinity, of the series

1 1 1 1
+ + +¼+ .
2
2
2
n 2 n - 1 n - 2 2 n - (n 1) 2
-
n 1
Solution: Here the rth term = å
2
-
=
r 1 n - (r 1) 2
Since it contains (r – 1), we consider its (r + l)th term i.e.,
n 1 n 1 1
the term å = å 2
2
r 0 n - r 2 r 0 n æ r ö
=
=
1 + ç ÷
è n ø

LOVELY PROFESSIONAL UNIVERSITY 259