Paper 4: Fundamentals of Business Mathematics & Statistic

3.88 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

If now h → 0 +, then n → ∞.

∴ ( )

1 n

0 n r 1

f x dx lim1 r.

→ ∞n n=

=^

(^)
Example 180:Evaluate :
10 10 10
n^10
lim1 2 3 ... n
→ ∞ n

+ + + +

Given expression =

10 10 10 10

n^10

lim1 1 2 3 ... n

→ ∞n n

(^) + + + +
(^)
10 10 10 10
n
lim1 1^23 ... n
→ ∞n n n n n

= + + + +

(^)
n^101111
10
n r 1^00
lim 1 r x dx x^1 0.^1
→ ∞n n= 11 11 11

=^ = =^ = − =

(^)
Example 181:Evaluate :
2 2 2 2
n^33333333
lim^123 ... n
→ ∞ 1 n 2 n 3 n n n

=^ + + + +^

(^) + + + + (^)
Given expression
2 2 2
n^333333
lim1 1.n 2 .n ... n n
→ ∞n 1 n 2 n n n

=^ + + +^

(^) + + + (^)
2 2 2
n^333
1 2 n
lim^1 n n ... n
n 1 2 n
(^1) n (^1) n (^1) n
→ ∞

=^ + + +^

(^) + + +
(^)
[dividing each term of numertator and denominator by n^3 ]
2
n 1 2
n r 1 3 0 3
r
lim n1 x dx.
n r 1 x
(^1) n
→ ∞ =
(^)
=^ =
+^ +
(^)
(^) [Put 1 + x (^3) = u. 3x (^2) dx = du and etc.]

( ) ( )

3 1

0

(^1) log 1 x (^1) log2 log1 log2. 1
3 3 3

=^ +^ = − =

(^)