A proof with a mistake ( or not ) Part 2

Number Theory Level 3

The Leibniz formula for π \pi states that 1 − 1 3 + 1 5 − 1 7 . . . . . . . = π 4 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} ....... = \frac{\pi}{4} or 4 ( 1 − 1 3 + 1 5 − 1 7 . . . . . . . ) = π . 4\left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} ....... \right) = \pi . The following procedure uses the Lebniz formula to prove that π = 4. \pi=4. Find the incorrect step.

Step 1 - Termwise, add 4 ( 1 3 − 1 5 + 1 7 . . . . . . . ) 4\left( \frac{1}{3} - \frac{1}{5} + \frac{1}{7} .......\right) to the LHS of the formula, and then subtract it at the end again, to keep the value constant at π \pi . Doing so, we get 4 ( 1 + 1 3 − 1 3 + 1 5 − 1 5 . . . . . . . . ) − 4 ( 1 3 − 1 5 + 1 7 . . . . . . . ) = π 4\left( 1 + \frac{1}{3} - \frac{1}{3} + \frac{1}{5} - \frac{1}{5} ........\right) - 4\left( \frac{1}{3} - \frac{1}{5} + \frac{1}{7} .......\right) = \pi

Step 2 - We can see that 1 3 − 1 5 + 1 7 . . . . . . . = − ( π 4 − 1 ) , \frac{1}{3} - \frac{1}{5} + \frac{1}{7} ....... = -\left(\frac{\pi}{4} - 1\right) , so 4 ( 1 ) − ( − ( π 4 − 1 ) = π 4(1) -(-(\frac{\pi}{4} - 1) = \pi

Step 3 - Simplifying,we get ( 4 − 1 + π 4 ) = π (4 - 1 + \frac{\pi}{4}) = \pi We subtract π 4 \frac{\pi}{4} from both sides to obtain 3 = 3 π 4 3 = \frac{3\pi}{4} But this means that π = 4 \pi = 4 !

Step 1 Step 3 There is no mistake and π = 4 \pi = 4 Step 2

Tan Li Xuan by Tan Li Xuan , 19, Malaysia

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4 solutions

Tan Li Xuan
May 6, 2014

The problem is in the second step.The fact that 1 3 − 1 5 + 1 7 . . . . . . . = − ( π 4 − 1 ) \frac{1}{3} - \frac{1}{5} + \frac{1}{7} ....... = -(\frac{\pi}{4} - 1) is true,however it should have been multiplied by 4 because the original expression was 4 ( 1 3 − 1 5 + 1 7 . . . . . . . ) 4(\frac{1}{3} - \frac{1}{5} + \frac{1}{7} .......) so it should be 4 ( 1 ) − 4 ( − ( π 4 − 1 ) ) 4(1) - 4(-(\frac{\pi}{4} - 1))

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Wow, I must admire you have brilliant posted a problem that really is convincingly true. For the first time I was pretty confused like "Hey, man, I studied math for this many years and this guy suddenly comes upfront saying that π = 4 \pi = 4 ?" Truly magnificent.

Umang Vasani - 7 years ago

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Thanks :) I actually came up with this in a weird way.When I have free time during class,I try and solve some problems on a piece of paper.I just happened to come up with this and I found the mistake while I was proofreading it.

Tan Li Xuan - 7 years ago
Anand Raj
Jun 23, 2014

4 should be common to both not only 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

that was a good question.It need a concentrated observation

0 Helpful 0 Interesting 0 Brilliant 0 Confused

4 is missing in step 2.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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