Evaluating Limit approaching Infinity.

Past sunday i gave a mathematics olympiad exam at IIT Bombay. I was asked a problem in which i had to find limit of problem :

limit x approaches infinity. \[\frac{1^{3} + 2^{3} + 3^{3} + … n^{3}}{n^{4}}\]

there are 2 ways of evaluating this limit.

1) by using the fact that $∑n^{3} = \frac{n^{2}(n+1)^{2}}{4}$

we get the value of limit as $\frac{1}{4}$

2) by separating the denominators and using the fact that limit of a sum is the sum of the limits.

$\frac{1^{3} + 2^{3} + 3^{3} + … n^{3}}{n^{4}}$

= $\frac{1^{3}}{n^{4}} + \frac{2^{3}}{n^{4}} + … + \frac{n^{3}}{n^{4}}$

And it turns out that the value of limit is 0.

I was confused at this while writing the exam. So I choosed $\frac{1}{4}$ by random.

And I am wondering if I did right decision or not. So I asked here. Please consider explaining me why we get two distinct answers and if my first or second approaches have any errors then please tell me. Thanks

#Calculus #Limits #Infinity #Summation #Olympiad

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

You chose correct answer. The 2nd method is wrong. It is expressed as sum of infinite infinitesimal quantities. It is something like saying 0 multiplied by infinity.

Reaber John - 7 years, 4 months ago

Cool explanation! The problem can be also evaluated using Riemann Sums which would be integration from $x=0$ to $x=1$ , the function $x^{3}$ .

Piyal De - 7 years, 4 months ago

Exactly.

Finn Hulse - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$