NMTC Final test 2015

Hello friends, help me in finding the value of

$\large\ a = 1 + \frac { 1 }{ { 2 }^{ 2 } } + \frac { 1 }{ { 3 }^{ 2 } } + ... + \frac { 1 }{ { 2015 }^{ 2 } }$ .

Do post solution.

#NumberTheory

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

Solution :

$\displaystyle \sum^{\infty}_{n=1}\dfrac{1}{n^{2}}=\dfrac{\pi^{2}}{6} \sim 1.64 \text{ and }\displaystyle \sum^{2015}_{n=1}\dfrac{1}{n^{2}}=a$

$\text{Simply by observation, we can conclude }\Rightarrow 1<a<1.64$

$\text{Therefore, }\lfloor a \rfloor=\boxed{1}$

Akshat Sharda - 5 years, 7 months ago

Thank for a solution. But please explain me how you got this:

$\large\ \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 2 } } = \frac { \pi ^{ 2 } }{ 6 } }$ ?

Priyanshu Mishra - 5 years, 7 months ago

See this.

Akshat Sharda - 5 years, 7 months ago

@Akshat Sharda – Thanks ,now i understood. However , how many points will you give for this problem?

Priyanshu Mishra - 5 years, 7 months ago

@Priyanshu Mishra – 75 of 400

Akshat Sharda - 5 years, 7 months ago

[a]=1

SARAN .P.S - 5 years, 7 months ago

Well,it is obvious if one knows the solution of the basel problem .However this is a much simpler question.Notice that the first term is 1,the next two terms are less than 2×1÷2^2 that is 1/2,the next four terms less than 1/4 and so on.Even if we did this til infinity,the expression would be less than 1+1/2+1/4.... which is 2.Hence the required value is 1.

Aditya Anand - 5 years, 7 months ago

that is now we have 1+....so on .consider 1 the next terms are decreasing values in decimals .so anyway if we add also we won't get more than 1 so the integral part of that expression i.e the[a] is 1...

SARAN .P.S - 5 years, 7 months ago

Question has asked for value of a, not floor function of a. So a will be equal to approx. 1.64 . Do note that the series is going smaller and smaller.

Silver Vice - 5 years, 6 months ago

In the question he didn't wrote about the integral value but in the original paper it was asked to find the integer part of $a$ .

Akshat Sharda - 5 years, 6 months ago

Yes, I forgot to write that.

By the way, how many questions you solved in the exam ?

Priyanshu Mishra - 5 years, 6 months ago

@Priyanshu Mishra – I didn't gave the exam :-(

Akshat Sharda - 5 years, 6 months ago

Ok, :) , integral value will be one only.

Silver Vice - 5 years, 6 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}`	$\boxed{123}$