1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + = ^2/6. What is the value of 1/1^2

Question

\frac{1}{1^2} + \frac{1}{2^2}+ \frac{1}{3^2} + \frac{1}{4^2} + … = \frac{π^2}{6}.  What is the value of  \frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + … ?

A.  \frac{π^2}{8}

B.  \frac{π^2}{2}

C.  π^2

D.  2π^2

E.  4π^2

ZeyadSerag · Accepted Answer

The first sequence is adding (1 / square of all integers)... The Second sequence is adding (1/square of all odd integers).

Clearly the second sequence should have a lower value than the first sequence.

Hence, going through the answers, only A has a lower value than (pi^2/6). Therefore, A is the correct answer.

kunalcvrce · Answer

\frac{1}{1^2} + \frac{1}{2^2}+ \frac{1}{3^2} + \frac{1}{4^2} + … = \frac{π^2}{6}. 
can also be written as:-

\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + …+\frac{1}{2^2}*(\frac{1}{1^2} + \frac{1}{2^2}+ \frac{1}{3^2} + \frac{1}{4^2} + …) = \frac{π^2}{6}.

\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + …+\frac{π^2}{6}*\frac{1}{2^2}= \frac{π^2}{6}. 
solving- we will get  \frac{π^2}{8}

MathRevolution · Answer

=>

Set  x = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + … . Then

\frac{π^2}{6} =\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + …

= (\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + … ) + (\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + … )

= (\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + … ) + (\frac{1}{(1*2)^2} + \frac{1}{(2*2)^2} + \frac{1}{(2*3)^2} + … )

= (\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + … ) + (\frac{1}{4})(\frac{1}{1})^2 + \frac{1}{2^2} + \frac{1}{3^2} + … )

=x+(\frac{1}{4})(\frac{π^2}{6})

So,  x= π^2/6-(1/4)(π^2/6)=(3/4)(π^2/6)=π^2/8

Therefore, the answer is A.
Answer: A

Sridip · Answer

Here we could use a easier method, as we know the first sequence is more than the next sequence. The value of first sequence will be more than the second sequence.

And Hence, the value should be lesser than pi/6 which is pi/8.

Abhibarua · Answer

Instead of lengthy calculations, this can be answered by looking at the options and using a bit of logic.

(1/1^2)+(1/3^2)+(1/5^2)+... will always be less than (1/1^2)+(1/2^2)+(1/3^2)+(1/4^2)+... = (π^2/6)

Only option A is less than π^2/6.

saynchalk · Answer

This is gonna be a stupid question, but I'm genuinely not aware of why:

why is it that the sequence with the squares of all integers is always less that the sequence with the squares of odd integers?

Bunuel · Answer

Because each term of the first sequence is more than (or equal to in the case of the first term) than each corresponding term of the second sequence.

harjas2222 · Answer

I require help with this question, can anyone explain it to me in a more breakdown manner step by step please.

Bunuel · Answer

Given:

\frac{1}{1^2} + \frac{1}{2^2}+ \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{π^2}{6}

Now group the odd-placed terms and even-placed terms:

(\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + ...)+(\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2} + \frac{1}{8^2} + ...) = \frac{π^2}{6}

Now factor out 1/2^2 from each term in the even-placed group:

(\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + ...)+(\frac{1}{2^2}*(\frac{1}{1^2} + \frac{1}{2^2}+ \frac{1}{3^2} + \frac{1}{4^2} + ...)) = \frac{π^2}{6}

Substitute the sum in the second parenthesis with π^2/6, thus we get:

(\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + ...)+(\frac{1}{2^2}*( \frac{π^2}{6})) = \frac{π^2}{6}

(\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + ...)= \frac{π^2}{6}- \frac{1}{2^2}* \frac{π^2}{6}

(\frac{1}{1^2}+ \frac{1}{3^2} + \frac{1}{5^2} + ...)= \frac{π^2}{8}

Answer: A.

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + = ^2/6. What is the value of 1/1^2

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + = ^2/6. What is the value of 1/1^2

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards