Product of Summation

Calculus Level 4

k = lim ⁡ n → ∞ ∑ r = 1 n r × ∑ r = 1 n 1 r ∑ r = 1 n r \large k = \lim_{n\rightarrow\infty}\frac{\displaystyle{\sum_{r=1}^n\sqrt{r}\times\sum_{r=1}^n\frac{1}{\sqrt{r}}}}{\displaystyle{\sum_{r=1}^n r}}

What is the value of 9 k 9k ?


The answer is 24.

View wiki
Prakash Chandra Rai by Prakash Chandra Rai , 24, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

......

Correct it as :we get 9k=24

For Any help in Limit as a Sum form visit, https://brilliant.org/discussions/thread/solving-limits-using-integration/?ref_id=522318

6 Helpful 0 Interesting 0 Brilliant 0 Confused

K is (8/3) and according to your answer the question incomplete .

Prince Kumar Maurya - 6 years, 5 months ago

Log in to reply

Really Sorry, I typed 9K But somehow It got Misprinted and I haven't seen this post for long time. Very very sorry and thank you

Prakash Chandra Rai - 6 years, 5 months ago

The denominator can just be written as 0.5 n 2 + 0.5 n 0.5n^2+0.5n

Kenny Lau - 5 years, 6 months ago

I think you have a mistake : at the numerator, the fraction before each sigma is 1 n \frac{1}{n} , should not contain a radical.

Hasan Kassim - 6 years, 3 months ago
Brock Brown
Dec 28, 2014

I used this bit of Python to approximate the value of k: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
 
from math import sqrt
infinity = 1000000
def f1():
    total = 0
    i = 1
    while i <= infinity:
        total += sqrt(i)
        i += 1
        yield total
def f2():
    total = 0
    i = 1
    while i <= infinity:
        total += float(1)/sqrt(i)
        i += 1
        yield total
def f3():
    total = 0
    i = 1
    while i <= infinity:
        total += i
        i += 1
        yield total
gen1, gen2, gen3 = f1(), f2(), f3()
for i in xrange(infinity):
    sum1 = next(gen1)
    sum2 = next(gen2)
    sum3 = next(gen3)
    ans = (sum1*sum2)/float(sum3)
print ans

This says that k is a little less than 2.6645, but I figure that due to precision errors the real value of k is 2 2/3. Therefore, 9*k is 24.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

This is a calculus problem, not a CS problem.

Joel Tan - 6 years, 5 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...